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## Calculus homework help

Comparing Two Salary Structures

PROBLEM SITUATION:

You have two job offers. Job 1 has an annual starting salary of \$60,000 with the expectation of a \$1000 raise each year. Job 2 has an annual starting salary of \$45,000 and expectation of a 7.5% annual raise.

(3pts) 1. Tables of future annual salaries can be used to begin to compare the two job offers. Copy the following tables into your report and complete them. Round to the nearest dollar.

TABLE 1 TABLE 2

## Job 1: Annual Salary

Job 2: Annual Salary

# of Years Worked

Annual Salary after t years

Successive

Differences

Successive Ratios

(3 decimal places)

# of Years Worked

Annual Salary after t years

Successive

Differences

Successive Ratios

(3 decimal places)

0

Salary of year 1 – salary of year 0

Salary or year 1/

salary of year 0

0

Salary of year 1 – salary of year 0

Salary or year 1/

salary of year 0

1

1

2

2

3

3

4

4

5

5

6

6

(1 pt ea) 2. Write the answers to questions a, b, and c using complete sentences. For question d, just write the function rule.

a. In which table are the successive differences constant?

b. What is that constant difference?

c. What type of function can be used to model this salary offer?

## Calculus homework help  Parametric Equations

Student Guide

Student Guide (continued)

Assignment Summary

For this assignment, you will represent hiking itineraries with parametric equations and use those equations to make predictions.

Background Information

Parametric equations are a set of functions defined in terms of a parameter. A parametric equation contains a horizontal and a vertical component that are both defined in terms of the parameter. The parameter can be eliminated to create a rectangular equation.

Assignment Instructions

For this project, you are expected to submit the assignment.

Step 1: Prepare for the performance task.

a) Read through the guide before you begin so you know the expectations for this assignment.

b) If there is anything that is not clear to you, be sure to ask your teacher.

c) If your word-processing program has an equation editor, you can insert your equations here. Otherwise, print this activity sheet and write your answers by hand.

Step 2: Complete Parts 1 and 2 in the Assignment section of this document.

c) Insert images or screenshots of graphs when needed. Be sure that all graphs or screenshots include appropriate information such as titles, labeled axes, etc.

d) Be sure to show all your work. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations.

e) Consider underlining and circling important components in the problems.

Step 3: Evaluate your project using this checklist.

If you can check each box below, you are ready to submit your project.

· Have you answered all questions in Part 1 and Part 2?

· Have you shown your work?

· Did you include an image or screenshot of a graph when requested?

· Are all your equations correct? Be sure to check your formatting carefully.

Step 4: Revise and submit your project.

a) If you were unable to check off all of the requirements on the checklist, go back and make sure that your project is complete. Save your project before submitting it.

b) Your teacher will give you further directions about how to submit your work. You may be asked to submit your responses through the virtual classroom, email it to your teacher, or print it and hand in a hard copy.

c) Congratulations! You have completed your project.

## Assignment

Part 1: Write parametric equations to represent a mountain hiking trail.

The table shows the estimated distances and elevation changes between a base camp and various locations along a popular mountain trail. The distance traveled along the trail is represented by x, the elevation is represented by y, and the time from the base camp to each location is represented by t.

 Distance from Base Camp Elevation Travel Time from Base Camp Base Camp 0 miles 6,990 feet 0 hours Camp I 4 miles 8,290 feet 10 hours Camp II 8.4 miles 10,875 feet 21 hours Camp III 11.4 miles 13,331 feet 28.5 hours Camp IV 14.8 miles 16,795 feet 37 hours Summit 17.2 miles 19,675 feet 43 hours

1. Use the data points about the distance and elevations of the camps to graph the data points where the x-coordinate is the total distance traveled from the base camp and the y-coordinate is the elevation. Use the grid below or include a screenshot of the data plotted from a calculator. (5 points) 2. Describe the curve in the graph by completing the steps.

a) Find a reasonable interval for the values of and explain its significance. (10 points)

b) Write a linear function, x(t), for the total distance in miles hiked in terms of total time hiking in hours using the data from the table. (5 points)

c) Write a quadratic function, y(t), for elevation in terms of the total time hiking in hours using the data in the table. Find a system of three equations and three unknowns to write the equation. (10 points)

Part 2: Write and use the rectangular form of the parametric equations.

1. Using the equations from Part 1, eliminate the parameter, and write the rectangular form of the equation. (10 points)

2. Use the rectangular equation to show the elevation that a group can expect to reach after hiking 8 miles. Then determine the time it will take the group to get to that point on the mountain. (6 points)

3. Use the rectangular equation to show approximately how far a group can expect to hike before reaching an elevation of 12,000 feet. (4 points)

## Calculus homework help

1. Calculate the following:

2. Find all the solutions for the following equations:

3. Given that:

Prove that:

4. Assuming the radius of convergence of the power series ∑ 𝑐 𝑧 is R. Calculate the
radius of convergence for the following power series:

5. Calculate the radius of convergence for the following power series:

6. Find the Holomorphic function f(z) if the following is given:

a .
b .
c .
d .
e .
f .

7. Prove that if 𝑓(𝑧) and 𝑓̅(𝑧) are Holomorphic functions in domain D, than 𝑓(𝑧) is a
constant function.

8. Prove that if 𝑓(𝑧) and 𝑔(𝑧) are Holomorphic functions in domain D, than

𝐼𝑚 𝑓(𝑧) + �̅�(𝑧) = 0 if and only if 𝑓(𝑧) − 𝑔(𝑧) = 𝐶 in domain D and C is a Real constant.

## Calculus homework help

Name: Last __________________ First ____________________________Section # _________

Signature Assignment Math 1316 Spring 2022

Due Date: Wednesday April 27, 2022 at 11:59 pm

[There will be a 20 point penalty assessed for turning in late]

The due date for this assignment is 11:59 pm on Wednesday April 27, 2022 at 11:59 pm. It is to be

uploaded as an online submission on Canvas. On Canvas go to Modules and then click on the

Then when you have finished the assignment click on submit assignment and a link will open up that

will let you upload your completed assignment. It is preferred that you upload it as a scanned pdf file

but if this is not possible you can upload it as a jpeg file by using your phone.. Use whatever method

works best for you – but make sure to have the completed assignment uploaded to Canvas no later

than 11:59 pm on April 27, 2022 .

1) Use the definition for the derivative 𝑓 ′(𝑥) = limℎ→𝑜
𝑓(𝑥+ℎ)−𝑓(𝑥)

to find the derivative of the

function f(x) = 3x
2
+ 10x – 5. You must show all of your work. You can take the derivative the quick

way to check your answer – but you must show how to compute the derivative using the above

formula to get credit for this problem. (10 points)

𝑓 ′(𝑥) = _______________________________________

2) Assume that the cost equation in dollars of producing x units of a product is given by the equation

C (𝒙)=𝟐𝟓𝟎𝟎+𝟏𝟎𝒙 and that the monthly demand equation for the product is given by the equation

𝑝 = 30 −
𝑥

1000
where x is the number of units demanded per month when the price charged is

p dollars. Use the above information to compute the monthly revenue equation for the product.

Then find the monthly profit equation for the above product and use it to compute the monthly

marginal profit function for this product. Finally use this to determine the profit and marginal profit

associated with a monthly production level of 8000 units. All work must be shown! (16 points)

Revenue Equation = R(x) = ____________________________

Profit Equation = P(x) = _______________________________

Profit (8000) = ______________________________________

MP(8000) = _________________________________________

3) Given 𝑦=(x+2)(2𝑥
2
+3)

3
find the equation of the tangent line to this function when x = 1.

First find the point on this function and the slope of the tangent line to this function when

x = 1. Next use these to find the equation of the tangent line to this function when x = 1.

Finally, put this equation in slope intercept form. All work must be shown!! (15 points)

Point on function when x = 1 is (1, _____)

Slope of tangent line when x = 1 is _____________

Equation of tangent line in slope intercept form is:

_______________________________________

4) A farmer needs to enclose by fences a rectangular field containing 392,000 square feet. One side
of the field lies along a river and one side lies along a road. The river is perpendicular ( at right

angles) to the road, He needs a more expensive fence on the sides next to the road and the river,

with a cheaper fence being used on the remaining 2 sides. The fence costs \$20 per foot along the

river, \$15 per foot along the road, and \$5 per foot on the remaining 2 sides to be fenced. Find the

dimensions he should use for the field to fulfill the above criteria and minimize the cost of fencing the

field. Also calculate the minimum cost possible for constructing a field that meets the above criteria?

Include a diagram of the field as part of your work. This diagram will help you determine the

correct cost function for constructing the fence. All work must be shown in order to get credit for

the problem!!!. (20 points)

The side next to the road must equal _____________ feet

The side next to the river must equal _____________ feet

The minimum cost of the fence is ___________________

5) Use integration by parts to evaluate the following integral :

∫ [3x
2
ln(x

3
)]dx. Identify the values that need to be used for u, du, dv, and v as well as

the final value for the integral. All work must be shown. (20 Points)

U = ___________________

du = ___________________

dv = ___________________

v = ____________________

∫ [3x
2
ln(x

3
)]dx = _____________________________

6a) Explain in your own words the meaning of the terms Consumer and Producer Surplus.

(4 Points)

6b) Given the demand equation for a product is p=D(𝑞)=200−0.3𝑞
2

and the supply function is

𝑝=S(𝑞)=0.2𝑞
2

find the equilibrium point for this product and use it to compute the value of the

Consumer and Producer Surplus associated with this product. All work must be shown. Round

answers off to two decimal places.

(15 points)

Equilibrium point _________________________

Consumer’s Surplus _______________________

Producer’s Surplus _______________________

GRAPH 1 GRAPH 2

Week Number

Items Sold

Week Number

Items Sold

## Calculus homework help

CombinedChainProductandQuotientRules

fix xcx 4 3 fix 4xe

fix 410 x fG SPEX

fix 2 073 130 fix x In Ite

f x 43125 2 014 fix 2 1 esh

fix xt3
3 2 17 fly ex In 3 2

fix 3 x NH 016 fix ÉIq
Fix 61 4 fix WEIL
fix 1372 180 fly

In Gtx
3X
Fx

fix F 190 fu entire
fix exfly Etta Inbox 2

For the given function, find the derivative.

## Calculus homework help

Name:

Study Guide for Exam 3

Show absolutely all work. Start each problem on a new page. Label them clearly.

The Problems
1. Find the general solutions of the given differential equations, initial value problem, and

system using whichever method that seems appropriate.

a) 𝑦(4) − 12𝑦’’’ + 53𝑦’’ − 100𝑦’ + 68𝑦 = 500𝑒2x

b) 𝑥3𝑦’’’− 7𝑥2𝑦’’ + 24𝑥𝑦’ − 34𝑦 = 50𝑥2, 𝑥 > 0

c) 8𝑦’’’ − 12𝑦’’ + 6𝑦’ − 𝑦 = 960𝑒x⁄2

d) 𝑥2𝑦’’ = 𝑦′(3𝑥 − 2𝑦’)

e) 4𝑦(𝑦’)2𝑦’’ = (𝑦’)4 + 3, assume x is the independent variable.

f) 2𝑥’ + 2𝑥 + 𝑦’ − 𝑦 = 𝑡 + 1, 𝑥’ + 3𝑥 + 𝑦’ + 𝑦 = 4𝑡 + 14. (Eliminate x first!)

2. Find the first six non-zero terms of a Taylor series solution to the initial value problem:

2𝑦3 − 4𝑒-x𝑦’ – 𝑦’’ = 𝑥2 sin 𝑥 , 𝑦(0) = −3, 𝑦’(0) = 1

## Calculus homework help

MATH 1325 QUIZ/PROJECT

separate sheets of paper then scan (or take picture) and submit to eCampus under the

same folder (Do not send them to my email).

1. A company’s total cost, in millions of dollars, is given by 𝐶(𝑡) = 120 − 80𝑒 −𝑡 where t =
time in years. Find the marginal cost when t = 4.

2. 𝑃(𝑥) = −𝑥 3 + 12𝑥 2 − 36𝑥 + 400, 𝑥 ≥ 3 is an approximation to the total profit (in
thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred

thousands of tires that must be sold to maximize profit.

3. Given the revenue and cost functions 𝑅 = 28𝑥 − 0.3𝑥2 and 𝐶 = 5𝑥 + 9, where 𝑥 is the daily
production, find the rate of change of profit with respect to time when 10 units are produced

and the rate of change of production is 4 units per day.

4. Find the producers’ surplus at a price level of �̅� = \$30 for the price-supply equation
𝑝 = 𝑆(𝑥) = 14 + 0.0004𝑥 2

Name:

## Study Guide for Exam 3

Show absolutely all work. Start each problem on a new page. Label them clearly.

## The Problems

1. Find the general solutions of the given differential equations, initial value problem, and system using whichever method that seems appropriate.

a) 𝑦(4) − 12𝑦’’’ + 53𝑦’’ − 100𝑦’ + 68𝑦 = 500𝑒2x

b) 𝑥3𝑦’’’− 7𝑥2𝑦’’ + 24𝑥𝑦’ − 34𝑦 = 50𝑥2, 𝑥 > 0

c) 8𝑦’’’ − 12𝑦’’ + 6𝑦’ − 𝑦 = 960𝑒x⁄2

d) 𝑥2𝑦’’ = 𝑦′(3𝑥 − 2𝑦’)

e) 4𝑦(𝑦’)2𝑦’’ = (𝑦’)4 + 3, assume x is the independent variable.

### f) 2𝑥’ + 2𝑥 + 𝑦’ − 𝑦 = 𝑡 + 1, 𝑥’ + 3𝑥 + 𝑦’ + 𝑦 = 4𝑡 + 14. (Eliminate x first!)

2. Find the first six non-zero terms of a Taylor series solution to the initial value problem:

2𝑦3 − 4𝑒-x𝑦’ – 𝑦’’ = 𝑥2 sin 𝑥 , 𝑦(0) = −3, 𝑦’(0) = 1

## Calculus homework help

FirstDerivTestforlocalExtrema oddsolutions

Fox 2 2 3 7 domain all reals

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3
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f x 4

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b

## Calculus homework help

From the attached problem set, submit #2, 10, 14, 16 for grading.

Follow the directions on the problem set. Use this problem set for more
practice – 4 problems only is not sufficient to prepare your for the rest of
the semester. Solutions to odds are available now. Solutions to even are
released after the due date.

completely and with the correct notation (including appropriate use of
“=”.)

Make sure everything is Handwritten on a white paper

Don’t forget to write my name and date on the paper
also scan the work and send it on a pdf form

Name: Yusuf Hussein

Date: 3/16/2022

## Calculus homework help

MATH 10283 – Applied Calculus Name: ___________

12.3 – Absolute Extrema
Find the absolute extrema (both x and y coordinates) of the function on the specified domain.
Show your work by listing all critical values and showing your test and conclusions.

1. f x( ) = x4 !18×2 +1 on !4,4[ ]

2. f x( ) = x3 ! 3×2 on !1,3[ ]

3. f x( ) = x3 !12x on 0,4[ ]

4. f x( ) =
1

x ! 2
on 0,1[ ]

5. f x( ) =
1

x2 +1
on 1,4[ ]

6. f x( ) =
1

x2 +1
on !1,1[ ]

7. f x( ) = 3x +1( )3 on !2,1[ ]

8. f x( ) =
3

x
2
+ 4

on !2,2[ ]

9. f x( ) =
1! x

3+ x
on 0,3[ ]

10. f x( ) =
x

x ! 2
on 3,5[ ]

11. f x( ) =
x

x
2
+ 2

on !1,4[ ]

12. f x( ) =
x
2
+1

x
on 0,!( )

13. f x( ) = 2x +
6

x
on 0,10( )

14. f x( ) = 3×2 ! 2 +
6

x
2

on !5,0( )

15. f x( ) =
x

e
x

on 0,3( )

16. f x( ) =
e
x

x !1
on 1,4( )

17. f x( ) = xex on !3,2( )

18. f x( ) = xe2!x
2

on !”,0( )

19. f x( ) = 8×12 ! x on 10,20( )

20. f x( ) = 6×23 ! 4x on 0,!( )

## Calculus homework help

the instructions on the problems set. Show and label your work as has
been demonstrated in class examples. You will be graded on notation,
algebra, calculus, and your diagrams/reasoning. You should do more
problems than are assigned in order to master this content.

Make sure everything is Handwritten on a white paper

Don’t forget to write my name and date on the paper
also scan the work and send it on a pdf form

Name: Yusuf Hussein

Date: 4/1/2022

## Calculus homework help

the instructions on the problems set. Show and label your work as has
been demonstrated in class examples. You will be graded on notation,
algebra, calculus, and your diagrams. You should do more problems
than are assigned in order to master this content.

Make sure everything is Handwritten on a white paper

Don’t forget to write my name and date on the paper
also scan the work and send it on a pdf form

Name: Yusuf Hussein

Date: 3/20/2022

## Calculus homework help

SUNY College of Environmental Science and Forestry

APM 105: Survey of Calculus and Its Applications I

Written HW 2

Chapter 1

Complete the following problems in a neat and organized manner, with exercises in sequential order.

Make sure to label each problem (for example: 3(c)) and show all work where appropriate.

If you have more than one sheet, be sure to staple your papers.

(1) Given the two functions: f(x) = x2 g(x) = 12 + 2x � x2
(a) Find the coordinates where f(x) and g(x) intersect.
(b) Find the distance between the intersection points.

(c) Find the point halfway between the intersection points (the midpoint).

(d) Write an equation of the line that contains the points of intersection.

(2) It costs a certain manufacturer \$2,500 to make 700 units of a product, and \$4,000 to make 1200 units of the product.

The manufacturer will sell the product at a fixed price of \$50 per unit.

(a) What is the Cost function, assuming the relationship is linear (C = mx + b)?
(b) What is the Revenue function?

(c) What is the Profit function?

(d) Find the number of units that need to be made and sold in order for the manufacturer to break even. Round

(3) When a pizza shop sells pizzas for \$15, they sell 700 pizzas per month. When the shop increased the price per pie

to \$18, they only sold 640 pizzas per month.

(a) What is the price (or demand) function, assuming the relationship is linear (p = mx + b)?
(b) What is the Revenue function?

(c) How many pizzas that need to be sold to reach a revenue of \$5,000? Round to the nearest whole pizza.

(d) Find the cost function if the overhead cost of running the shop is \$4,000 per month, and the cost for ingredients

to make each pizza is \$2.50.

(e) Find the Profit function.

(f) Find the production level (number of pizzas made and sold) that would cause the shop to break even. Round

to the nearest whole pizza.

For the functions in (4) to (7):
(a) Find the domain, in interval notation.
(b) Find the y-intercept.
(c) Find the x-intercept.

(d) Evaluate its di↵erence quotient
f(x+h)�f(x)

h . Fully simplify the result to the point of canceling out the original h from the denominator.

(4) f(x) = 5x � 2 (5) f(x) = x2 � 4x (6) f(x) =
p
3x + 2

(7) f(x) =
2

x

(8) Let f(x) =
x2 + 4x + 4

x2 � 4
.

(a) Find any discontinuities of f(x). For each, classify their type (removable or non-removable).
(b) Find lim

x!�2
f(x).

(c) Find lim
x!2

f(x).

(9) Let f(x) =
2 � x

x2 � 8x + 12
.

(a) Find any discontinuities of f(x). For each, classify their type (removable or non-removable).
(b) Find lim

x!2
f(x).

(c) Find lim
x!6

f(x).

I

.

(10) Sketch an example of a function f(x) with the following properties.
(a) f(3) exists but lim

x!3
f(x) does not exist.

(b) lim
x!3

f(x) exists but f(3) does not exist.

(11) Let g(x) =

p
x + 1 � 1

x
.

(a) Find the domain of the function g(x) in interval notation.
(b) Complete the table below, accurate to five decimal places. Use the result to estimate the limit in the center.

x �0.1 �0.01 �0.001 lim
x! 0

p
x + 1 � 1

x
0.001 0.01 0.1

g(x)

(c) Find the limit lim
x!0

p
x + 1 � 1

x
analytically (algebraically, by hand). Show your work.

(12) Evaluate each of the limits algebraically, or show it does not exist.

(a) lim
x!2�

5 � x
x + 2

(b) lim
x!16

p
x � 4

16 � x

(c) lim
x!0

1
4+x �

1
4

x

(d) lim
h!0

p
x + h �

p
x

h

(e) lim
x!�3+

2

x + 3

(f) lim
x!2�

x � 2
|x � 2|

The graph of y = f(x) is shown below. Use it to answer questions (13) to (15).

5

-2

-1

1

2

3

4

y

x

-4 -3 -2 -1 1 2 3 4 5 6 7 8 9

(13) Visually determine the following limits and function values.

a) lim
x!�3+

f(x) b) lim
x!�3�

f(x) c) lim
x!�3

f(x) d) lim
x!�2+

f(x) e) lim
x!�2�

f(x) f) lim
x!�2

f(x)

g) lim
x!�1+

f(x) h) lim
x!�1�

f(x) i) lim
x!�1

f(x) j) lim
x!0+

f(x) k) lim
x!0�

f(x) l) lim
x!0

f(x)

m) lim
x!2+

f(x) n) lim
x!2�

f(x) o) lim
x!2

f(x) p) lim
x!4+

f(x) q) lim
x!4�

f(x) r) lim
x!4

f(x)

s) lim
x!5+

f(x) t) lim
x!5�

f(x) u) lim
x!5

f(x) v) f(0) w) f(2) x) f(4)

(14) Visually identify the x-values where f(x) is discontinuous. State the continuity condition each point violates.

(15) What new value should be assigned to f(2) to make the function continuous at x = 2?

(16) Let f(x) =

x2 � 1, x  0
4x + 1, x > 0

. Sketch the graph of f(x) and discuss whether or not it is continuous.

For the functions in (17) to (19): Algebraically find any discontinuities, and state their type (removable or non-removable).

(17) f(x) =
x + 2

x2 � 4 (18) f(x) =
x4 � x3 � 6×2

x3 + 5×2 + 6x
(19) f(x) =

4×2 � 1
6×2 � 11x + 4

## Calculus homework help

SecondDerivativetestforlocalExtrema Oddsolutions

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## Calculus homework help

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## Calculus homework help

instructions on the problems set. Show and label your work as has
been demonstrated in class examples. You should use the second
derivative test for local extrema where applicable and the explain why if
you are unable to use it. You will be graded on notation, algebra,
calculus, and your diagrams. You should do more problems than are
assigned in order to master this content.

Make sure everything is Handwritten on a white paper

Don’t forget to write my name and date on the paper
also scan the work and send it on a pdf form

Name: Yusuf Hussein

Date: 3/20/2022

## Calculus homework help

MATHS 253 Semester Test 13 April 2022

This test is marked out of 100.

1. [20 points] Let V = R2[x] be the vector space of all real polynomials of degree at most 2 in the
variable x, and let D : V → V be the linear operator defined by

D( f )(x) = f (x) + (x − 1) f ′(x) + x f ′(x − 1) (1)

(you do not need to prove that D is a linear operator).

(a) [10 points] Let B = {1, x, x2}. Find [D]B, the matrix of D relative to the basis B.
(b) [10 points] Prove that D is invertible. Find a polynomial f ∈ R2[x] such that

D( f )(x) = 5×2 + 2x + 1. (2)

2. [20 points] Let

A =



2 0 1 0 0
0 2 0 0 0
0 0 2 0 0
0 0 0 1 −1
0 0 0 0 1



(a) [5 points] Write down the characteristic polynomial of A. What are the eigenvalues of A?
(b) [15 points] For each eigenvalue of A, find its geometric and algebraic multiplicities. Is A

diagonalizable?

3. [20 points] Let V = C∞
R
[−1, 1], the space of infinitely-differentiable functions f : [−1, 1] → R.

(a) [5 points] Let W be the subset of V of functions f which satisfy f (−1) = f (1) = 0. Prove
that W is a subspace of of V.

(b) [12 points] Let U be the subspace of W of functions f which also satisfy

dn f
dxn

(−1) =
dn f
dxn

(1) = 0 for all n ∈ N

(you do not need to prove that U is a subspace of W). Define an inner product on U by

( f , g) =
∫ 1
−1

f (t)g(t) dt

(you do not need to prove that this is an inner product). Let D : U → U be the linear
operator defined by

D( f )(x) =
d

dx
f (x).

(you do not need to prove that D is a linear operator). Prove that D satisfies D∗ = −D.
Such an operator is called skew-Hermitian.

Hint. Use integration by parts.

(c) [3 points] Using part (b), show that if f ∈ U, then f is orthogonal to its derivative.

QUESTIONS CONTINUE ON NEXT PAGE

4. [20 points] Let W ⊆ R3 be the plane given by x − 2y + z = 0.

(a) [10 points] Starting with the basis

B =



 11

1

 ,

 32

1



for W, run the Gram-Schmidt algorithm to find an orthonormal basis for W. (You do not
need to prove that B is a basis).

(b) [10 points] Find the matrix of orthogonal projection onto W and hence find the closest
vector v ∈ W to x = (1, 0, 0)T .

5. [20 points] Let

A =

 −1 0 00 −3 1

0 1 −3

(a) [2 points] Write down the quadratic form Q(x1, x2, x3) which A represents.

(b) [6 points] Is Q positive definite, negative definite, or indefinite? Justify your answer.

(c) [10 points] Find a basis of R3 in which Q has no cross-terms. Write down the quadratic
form Q(y1, y2, y3) relative to this basis.

(d) [2 points] Classify the quadric surface Q(x1, x2, x3) = −1.

## Calculus homework help

2

MATH

Choose ONE problem. Answer the problem questions using a complete sentence incorporating the language and context of each problem as appropriate. Show all the steps using proper notation to complete these problems as were presented in class. That means show first and second derivative tests as appropriate. Show your work when forming equations. Make sure your work is neat, answers are legible, problem is neatly presented free of scratch outs. Outline the steps you took to complete the signature assignment.

After you have competed the signature assignment problem, you will respond to the following prompts and write a
reflection
(5-10 sentences) utilizing the Signature Assignment Prompts to summarize the assignment.

Signature Assignment Reflection Prompts

1. What impact did the Signature Assignment have on your understanding of calculus?

2. What new knowledge did you gain about your calculus topic? Briefly explain.

3. What did you learn by creating the signature assignment?

4. What challenges did you face completing the assignment?

5. How will calculus be useful to you in the real world?

Problems: (Choose one)

1. The price-demand equation for a GPS device is where is the demand and is the price in dollars.

A. Find the production level that produces the maximum revenue.

B. Find the price per unit that produces the maximum revenue.

C. What is the maximum revenue?

D. If the GPS device cost the store \$220 each, find the price to the nearest cent that maximizes the profit.

E. What is the maximum profit to the nearest dollar?

G. Write the signature assignment reflection using the assignment prompts.

2. At a price of \$8 per ticket, a musical theatre group can fill every seat in the theatre, which has a capacity of 1500. For every additional dollar charged, the number of people buying tickets decreases by 75.

A. What ticket price maximizes revenue?

B. What is the maximum revenue?

D. Write a signature assignment reflection using the assignment prompts.

3. The concentration of a drug in the blood stream , in milligrams per cubic meter any time , in minutes is described by the equation

Where corresponds to the time at which the drug was swallowed.

A. Determine how long it takes the drug to reach its maximum concentration.

B. What is the maximum concentration?

D. Write a signature assignment reflection using the assignment prompts.

4. A motel finds that it can rent 200 rooms per day if it charges \$80 per room. For each \$2 increase in rental rate, two fewer rooms will be rented per day.

A. If each rented room costs \$8 to service per day, how much should the management charge for each room to maximize gross profit?

B. What is the maximum gross profit?

D. Write a signature assignment reflection using the assignment prompts.

5. Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be

It also determines that the total cost of producing x suits is given by

A. Find the maximum revenue.

B. How many suits must be sold to maximize revenue?

C. How many suits must the company produce and sell in order to maximize profit?

D. Find the maximum profit.

E. What price per suit must be charged in order to maximize profit?

F. If the government decides to tax the company the \$10 for each suit it produces, how many suits should the company produce to maximize its profit? What is the maximum profit with the government tax? How much should the company charge for each suit?

H. Write a signature assignment reflection using the assignment prompts.

## Calculus homework help

MATHS 253, Algebra. Lecture 16

4 April 2022

Colour code: Red

Preparation: Go through the text:
Textbook: §5.5 pp. 432 – 440.
Lecture Notes: Sec. 6.3.

1 / 31

Conic Sections
Geometrically conic sections are defined as an intersection of a plane with a right
circular cone.

These are parabola, ellipse and hyperbola and shown below:

2 / 31

The equation of the conic section

Algebraically, with the help of calculus it is not too difficult to show that in any
coordinate system in the intersecting plane the equation of the curve will be

ax 21 + 2bx1x2 + cx
2
2 + dx1 + ex2 + f = 0

for some real numbers a, b, c, d, e, f .

This can be written as
xT Ax + Mx + f = 0,

where A =
[

a b
b c

]
, M =

[
d e

]
, x =

[
x1
x2

]
.

and
ax 21 + 2bx1x2 + cx

2
2 = x

T Ax.

3 / 31

Parabola
y = ax 2 or x = ay 2. The parabola is along the axis corresponding to the variable
not being squared.

This is x = ay 2.
4 / 31

Ellipse

x 21
a2

+
x 22
b2

= 1.

An ellipse looks like an elongated circle. It extends in the x1-axis direction from −a
to a and in the x2-axis direction from −b to b. For example,

x 2

4
+

y 2

9
= 1.

Points (0,±3) are
called vertices of
the ellipse.

5 / 31

Hyperbola
x 21
a2

x 22
b2

= 1 or
x 22
a2

x 21
b2

= 1.

The hyperbola never crosses the axis corresponding to the variable having the
negative sign.

−x
2

4 +
y 2
9 = 1

x 2
4 −

y 2
9 = 1 6 / 31

Example
Classify the curve

3x 2 − 4xy + 3y 2 = 5

(i.e., say if it is parabola, ellipse or hyperbola, etc.), calculate its parameters and
draw the picture.

We need to diagonalise this quadratic form. As∣∣∣∣ 3 −λ −2−2 3 −λ
∣∣∣∣ = λ2 − 6λ + 5 = (λ− 1)(λ− 5)

in the coordinates relative to the basis of principal axes the equation of the curve
will be

z2 + 5t 2 = 5 or
z2

5
+

t 2

1
= 1.

7 / 31

So this is an ellipse whose basis B of principal axes consists of the normalised
eigenvectors

v1 =
1

2

[
1
1

]
, v2 =

1

2

[
−1

1

]
.

The vertices of this ellipse in the new coordinate system will be (±

5, 0). The
vertices in the old coordinates will be

PE←B

[ √
5
0

]
=

1

2

[
1 −1
1 1

][ √
5
0

]
=

[ √
5/2√
5/2

]

PE←B

[

5
0

]
=

1

2

[
1 −1
1 1

][

5
0

]
=

[

5/2

5/2

]
,

hence the vertices are at (

5√
2
,

5√
2
) and (−

5√
2
,−

5√
2
).

8 / 31

Geometric property of a parabola
A parabola is the set of points in a plane that are equidistant from a fixed point F
(called the focus) and a fixed line (called the directrix).

|PF| =

x 2 + (y − p)2 = |y + p| (the distance to the line)

x 2 + (y − p)2 = (y + p)2

x 2 = 4py.
9 / 31

Geometric property of an ellipse
An ellipse is the set of points in a plane the sum of whose distances from two fixed
points F1 and F2 (called the foci) is a constant.

|PF1|+ |PF2| = 2a√
(x + c)2 + y 2 +

(x − c)2 + y 2 = 2a

(a2 − c2)x 2 + a2y 2 = a2(a2 − c2).
x 2

a2
+

y 2

a2 − c2
= 1.

10 / 31

Geometric property of a hyperbola
An hyperbola is the set of points in a plane the difference of whose distances from
two fixed points F1 and F2 (called the foci) is a constant, i.e., |PF1|− |PF2| = ±2a:

x 2

a2

y 2

b2
= 1 where b2 = c2 − a2.

As x and y get large x
2

a2
= 1 + y

2

b2
=⇒±xa =

y
b . and the graph approaches the

asymptotes y = ±ba x .
If hyperbola is rotated 90◦ the minus shifts from y 2 to x 2.

11 / 31

Getting rid of linear terms

We shift conic h units to the right and k units up by taking the standard equation
(of parabola, ellipse or hyperbola) and replacing x and y by x − h and y − k .

Sketch the conic

9x 2 − 4y 2 − 72x + 8y + 176

After completing squares and divid-
ing by 36 we get

(x − 4)2

4
+

(y − 1)2

9
= 1.

12 / 31

Getting rid of linear terms not always possible

Sketch the conic and find the vertex and the focus:

y 2 − 8y = 6x − 16.

Solution: After completing squares we get

(y − 4)2 = 6x.

Hence the vertex is at (0, 4) and (since 4p = 6) the focus is at (1.5, 4).

When there is no square you cannot get rid of the corresponding linear term.

13 / 31

Getting rid of cross-product terms
The presence of cross-product and linear terms means that it has been rotated
and translated out of standard position, e.g.,

5x 21 − 4x1x2 + 8x
2
2 + 4

5×1 − 16

5×2 + 4 = 0

is a shifted and rotated ellipse

14 / 31

Example continued
We must choose new axes which coincide with the principal axes of the

5x 21 − 4x1x2 + 8x
2
2 = [x1 x2]

[
5 −2
−2 8

][
x1
x2

]
= xT Ax.

and then adjust the linear form

4

5×1 − 16

5×2 = [ 4

5,−16

5 ]
[

x1
x2

]
= Mx.

The characteristic equation of A is

det(A −λI) = (5 −λ)(8 −λ)− 4 = λ2 − 13λ + 36 = (λ− 4)(λ− 9),

that is, the eigenvalues are λ1 = 4 and λ2 = 9.

The orthonormal bases of one-dimensional eigenspaces are v1 =
1√
5
(2, 1)T and

v2 =
1√
5
(−1, 2)T , respectively.

15 / 31

Example continued
Let F = {v1, v2} be the corresponding basis of R2. Then we can write the change
of basis matrix

P = PE←F =
1

5

[
2 −1
1 2

]
.

This is the matrix of anticlockwise rotation through an angle arctan 1/2 which is
approximately 26.6◦. Then we have

PT AP = D =
[

4 0
0 9

]
,

and we express the vector of new coordinates y = [x]F of x by substituting
x = [x]E = PE←F [x]F = Py. Then we obtain

yT Dy + MPy + f = 0, where MP = [−8,−36]
or

4y 21 + 9y
2
2 − 8y1 − 36y2 + 4 = 0.

We have gotten rid of cross-products.
16 / 31

Example continued
However, the origin is still shifted. Completing the squares:

4(y1 − 1)2 + 9(y2 − 2)2 = 36 or
(y1 − 1)2

9
+

(y2 − 2)2

4
= 1.

This shows that in the new coordinates the curve is a shifted ellipse with one axis
having length 3 and another length 2.

17 / 31

This degenerate quadric surface is called elliptic cylinder.

x 2

a2
+

y 2

b2
= 1

It appears when one eigenvalue (the third one) is zero and the coefficient of z is
also zero.

18 / 31

This degenerate quadric surface is called hyperbolic cylinder.

x 2

a2

y 2

b2
= 1

It appears when one eigenvalue (the third one) is zero and the coefficient of z is
also zero.

19 / 31

This degenerate quadric surface is called parabolic cylinder.

x 2 + 2ay = 0

It appears when two eigenvalues (the second and the third one) are zero and the
coefficient of z is also zero.

20 / 31

This quadric surface is called ellipsoid.

x 2

a2
+

y 2

b2
+

z2

c2
= 1

It has semiaxes a, b, c, respectively.
21 / 31

Quadric surfaces. Hyperboloids of one sheet
This quadric surface is called hyperboloid of one sheet.

x 2

a2
+

y 2

b2

z2

c2
= 1

It has semiaxes a and b, respectively.
22 / 31

Quadric surfaces. Hyperboloids of two sheets
This quadric surface is called hyperboloid of two sheets.

x 2

a2

y 2

b2
+

z2

c2
= 1

A good rule to remember: one minus – one sheet, two minuses – two sheets.
23 / 31

This degenerate quadric surface is called circular cone.

x 2

a2
+

y 2

b2

z2

c2
= 0

It appears when one eigenvalue is negative and the constant term is zero.
24 / 31

Example

Let Q : R3 → R be the quadratic form, given by

Q(x) = 2x 21 + 2x
2
2 + 2x

2
3 + 2x1x2 + 2x1x3 + 2x2x3.

• Write down the matrix of this form in the standard basis E .
• Diagonalise the form Q, that is find an orthonormal basis F in which the form

will be given by the formula

Q(x) = λ1y
2
1 + λ2y

2
2 + λ3y

2
3 ,

[x]F = (y1, y2, y3)T ? Write down this basis and this formula, with all
coefficients given numerically.

• Determine the type of the surface Q(x) = 1.

25 / 31

Example continues
The matrix of this form is

A =

 2 1 11 2 1

1 1 2

 .

The characteristic polynomial of this matrix is equal to

det(A −λI) =

∣∣∣∣∣∣
2 −λ 1 1

1 2 −λ 1
1 1 2 −λ

∣∣∣∣∣∣ =
∣∣∣∣∣∣

1 −λ −1 + λ 0
0 1 −λ −1 + λ
1 1 2 −λ

∣∣∣∣∣∣ =
(we subtract the third row from the first and the second)

(λ− 1)2
∣∣∣∣∣∣
−1 1 0

0 −1 1
1 1 2 −λ

∣∣∣∣∣∣ = − (λ− 1)2(λ− 4).
Therefore there are two distinct eigenvalues: λ1,2 = 1, λ3 = 4; one of them 1 is of
algebraic multiplicity 2.

26 / 31

Example continues
Let us find the eigenvectors. Take λ1,2 = 1 first.

A −λ1I = A − I =

 1 1 11 1 1

1 1 1

 →

 1 1 10 0 0

0 0 0

 .

This row reduced echelon form corresponds to the system

x1 + x2 + x3 = 0,

whose solutions are spanned by the vectors f1 = (−1, 1, 0)T and f2 = (−1, 0, 1)T .
Orthogonalizing we get:

g1 = f1 = (−1, 1, 0)T ,

g2 = f2 −
f2 · g1
|g1|2

g1 = (−1, 0, 1)T −
1
2
(−1, 1, 0)T =

(

1
2
,−

1
2
, 1
)T

.

27 / 31

Example continues

After normalization we have

v1 =
1

2
(−1, 1, 0)T ,

v2 =
1

6
(1, 1,−2)T .

The third eigenvector belonging to λ1 = 4 can be found as a unit vector orthogonal
to v1 and v2. It can be taken as

v3 =
1

3
(1, 1, 1)T .

28 / 31

Example continues

In this basis F = {v1, v2, v3} the form is presented by

Q(v) = z21 + z
2
2 + 4z

2
3,

[v]F = (z1, z2, z3)T .

The surface Q(v) = 1 is an ellipsoid whose two semiaxes are equal to 1 (so that it

is an ellipsoid of revolution) and the third semiaxis is equal to
1
2

.

29 / 31

To Summarise

As a result of today’s lecture (together with your further study) you should
• be able to determine whether a conic section is a parabola, hyperbola or

ellipse and determine its position;
• be able to determine the type of a quadric.

30 / 31

What to do now

Consolidation: Try exercises:
Textbook: §5.5, pp. 441–442: # 25,27,31,33,

35,37,39,61,63,65,67,69,73,75,79.
Lecture Notes: Exercises of Sec. 6.3.

Next topic: Positive definite quadratic forms.

Colour code: Red

Preparation: Go through the text:
Textbook: §5.5; pp. 429–432.
Lecture Notes: Sec. 6.4–6.5.

31 / 31

## 1241aea110249b5359051ed84d5a6606.pdf

MATHS 253, Lecture 8

16 March 2022

Today’s topic: The Cayley-Hamilton Theorem, Minimal Polynomials
Colour code: Red.
Lecture Notes: Sec. 3.1–3.3

Textbook: §6.1

1 / 18

Algebraically Closed Fields

A field F is called algebraically closed if every polynomial f (x) ∈ F [x] of degree at
least 1 can be written as a product of linear factors:

f (x) = c(x − r1)(x − r2) · · ·(x − rn)

for some c, r1, r2, . . . , rn ∈ F .

Example 1

1. C is algebraically closed.
2. R is not algebraically closed (e.g. f (x) = x 2 + 1 cannot be written as a

product of linear factors over R).

For any field F , there exists another field E ⊇ F such that E is algebraically closed.

2 / 18

Uniqueness of Linear Factors Decomposition
Theorem 2
Let F be an algebraically closed field, and let f ∈ F [x]. Then the decomposition of
f into linear factors is unique (up to the order of the factors).

Proof.
Suppose we have two factorizations

f (x) = c(x − r1) · · ·(x − rn) = c′(x − r ′1) · · ·(x − r

n′).

We have
1. n = n′, since otherwise the two expressions have different degrees.
2. c = c′, since otherwise the two expressions have different x n coefficients.
3. If some rj 6∈ {r ′1, r

2, . . . , r

n′}, then the first expression says f (rj ) = 0, while the

second says f (rj ) 6= 0. So {r1, r2, . . . , rn}⊆{r ′1, r

2, . . . , r

n′} and vice versa.

3 / 18

Polynomials of Operators

If T : V → V is a linear operator on an F -vector space V and f ∈ F [x], we define
the operator f (T ) as follows: write f (x) = a`x` + a`−1x`−1 + · · ·+ a1x + a0; then

f (T )(v) = a`T
`v + a`−1T

`−1v + · · ·+ a1T v + a0v.

Notice that for any basis B, we have

[f (T )(v)]B = [a`T
`v + a`−1T

`−1v + · · ·+ a1T v + a0v]B
= [a`T

` + a`−1T
`−1 + · · ·+ a1T + a0I]B[v]B

=
(

a`[T ]
`
B + a`−1[T ]

`−1
B + · · ·+ a1[T ]B + a0[I]B

)
[v]B

= f ([T ]B)[v]B.

so that [f (T )]B = f ([T ]B).

4 / 18

Annihilating Polynomials

Definition 3
Let V be an n-dimensional vector space over F , and let T : V → V be a linear
operator. A polynomial f ∈ F [x] is called an annihilating polynomial for T if

f (T ) = 0 that is, if f (T )(v) = 0 for all v ∈ V .

Definition 4
The minimal polynomial µT of an operator T is the polynomial of lowest degree,
with leading coefficient 1, such that µT (T ) = 0.

5 / 18

Example: Annihilating Polynomials
Example 5
Let V = C3, and let T : V → V have standard matrix

A =

 1 0 10 1 1

0 0 2

Then f (x) = x 3 − 4x 2 + 5x − 2 and g(x) = x 2 − 3x + 2 are annihilating
polynomials for T .
How do we know this? We compute

f (A) = g(A) =

 0 0 00 0 0

0 0 0

so f (T )(v) = f (A)v = 0 and g(T )(v) = g(A)v = 0 for any v ∈ V .
6 / 18

On Annihilating Polynomials
Lemma 6
Let V be an n-dimensional F vector space with basis B = {v1, v2, . . . , vn}, and let
T : V → V be a linear operator. A polynomial f ∈ F [x] annihilates T if and only if

f ([T ]B) =



0 0 . . . 0
0 0 . . . 0

. . .

0 0 . . . 0

 i.e., f annihilates [T ]B .

Proof.
If f ([T ]B) = 0, then [f (T )(v)]B = f ([T ]B)[v]B = B for all v, so T is annhilated by f .
If f ([T ]B) 6= 0, suppose its i th column is ai 6= 0. Then

[f (T )(vi)]B = f ([T ]B)[vi ]B = f ([T ]B)ei = ai 6= 0

so T is not annhilated by f .
7 / 18

Existence of Annihilating Polynomials

Lemma 7
Let V be an n-dimensional F -vector space and let T : V → V be a linear operator.
Then there is a non-zero polynomial f ∈ F [x] of degree at most n2 which
annihilates T .

Proof.
Let A be the standard matrix of T . It suffices to show that there is f ∈ F [x] of
degree at most n2 such that f (A) = 0.
Consider the collection {I, A, A2, . . . , An

2
}∈ Fn×n. Since dim Fn×n = n2, this set is

linearly dependent. So there are b0, b1, . . . , bn2 such that

b0 + b1A + b2A
2 + . . . + bn2 A

n2 = 0.

We can take f (x) = b0 + b1x + . . . + bn2 x
n2 .

8 / 18

Cayley-Hamilton Theorem

It turns out we can do much better:

Theorem 8
Let V be an n-dimensional F -vector space and let T : V → V be a linear operator.
Then the characteristic polynomial pT annihilates T

Proof.
The proof is quite technical. See course notes Sections 3.2–3.3 for details.
How is this better? Two ways:

1. deg pT (x) = n � n2.
2. This way is constructive (gives a method to find an annihilating polynomial).

9 / 18

Example 9
Let T : R3 → R3 have standard matrix

A =

 1 2 22 1 −2
−2 2 5

Find a polynomial f ∈ R3[x] which annihilates A.

10 / 18

By the Cayley-Hamilton theorem we can take f (x) = pA(x). We compute

pA(λ) = det(A −λI3) =

∣∣∣∣∣∣
1 −λ 2 2

2 1 −λ −2
−2 2 5 −λ

∣∣∣∣∣∣ =
∣∣∣∣∣∣

1 −λ 2 2
2 1 −λ −2
0 3 −λ 3 −λ

∣∣∣∣∣∣
= (3 −λ)

∣∣∣∣∣∣
1 −λ 2 2

2 1 −λ −2
0 1 1

∣∣∣∣∣∣ =
∣∣∣∣∣∣

1 −λ 0 0
2 1 −λ −2
0 1 1

∣∣∣∣∣∣
= (1 −λ)(3 −λ)

∣∣∣∣ 1 −λ −21 1
∣∣∣∣ = (1 −λ)(3 −λ)2.

So we can take f (x) = (1 − x)(3 − x)2.

11 / 18

Finding the Minimal Polynomial

Perhaps surprisingly, the Cayley-Hamilton theorem gives us a reasonably-efficient
method to find the minimal polynomial of an operator.

Lemma 10
Let T : V → V be a linear operator, and suppose f ∈ F [x] annihilates T . Then
µT (x)|f (x).

Corollary 11
Let T : V → V be a linear operator. Then µT (x)|pT (x).

12 / 18

Proof (of Lemma).
Let A be the standard matrix of T .

Write f (x) = µT (x)q(x) + r(x) with deg r < deg µT .

We have 0 = f (A) = µT (A)q(A) + r(A) = r(A).

If r(x) 6= 0, then r annihilates A and has degree smaller than µT . That can’t
happen.

So r(x) = 0, so that f (x) = µT (x)q(x); that is, µT (x)|f (x).

Proof (of Corollary).
Take f (x) = pT (x) and apply the Cayley-Hamilton theorem and the Lemma.

13 / 18

Finding the Minimal Polynomial

We can use this to find the minimal polynomial of T as follows:
1. Write down the standard matrix A of T .
2. Compute pT (x) (= pA(x))
3. Write down the divisors {f1(x), f2(x), . . . , f`(x)} of pT (x).
4. Find the smallest degree fj (x) which satisfies fj (A) = 0. That’s µT (x).

14 / 18

Example 12
Find the minimal polynomial of T : R3 → R3, which has standard matrix

A =

 1 2 22 1 −2
−2 2 5

 .

We already know pT (λ) = (1 −λ)(3 −λ)2. The polynomials that divide pT are:
f1(λ) = 1, f2(λ) = 1 −λ, f3(λ) = 3 −λ,
f4(λ) = (1 −λ)(3 −λ), f5(λ) = (3 −λ)2, f6(λ) = (1 −λ)(3 −λ)2.

We test them:

f1(A) =

 1 0 00 1 0

0 0 1

 f2(A) =

 0 −2 −2−2 0 2

2 −2 −4

 f3(A) =

 2 −2 −2−2 2 2

2 −2 −2

15 / 18

More testing:

f4(A) =

 0 −2 −2−2 0 2

2 −2 −4

 2 −2 −2−2 2 2

2 −2 −2

 =

 0 0 00 0 0

0 0 0

f5(A) =

 2 −2 −2−2 2 2

2 −2 −2

2 =

 4 −4 −4−4 4 4

4 −4 −4

f6(A) =

 0 −2 −2−2 0 2

2 −2 −4

 2 −2 −2−2 2 2

2 −2 −2

2 =

 0 0 00 0 0

0 0 0

So f4, f6 annihilate A. f4 has the smaller degree, so µT (x) = f4(x) = x 2 − 4x + 3.

16 / 18

To Summarise

As a result of today’s lecture (together with your further study) you should
• Know what it means for a field to be algebraically closed.
• Know how to compute the evaluation of a polynomial at a matrix/operator.
• Know the definitions of annihilating polynomial and minimal polynomial.
• Be able to state the Cayley-Hamilton theorem.
• Be able to find the minimal polynomial of a matrix/operator.

17 / 18

What to do now

Consolidation: Lecture Notes: Exercises of Sec. 3.3;

Textbook: §4.3, Exercises 33–38
Next topic: Jordan Normal Form

Color code: Red.

Preparation: Go through the text:
Lecture Notes: Sec. 3.4–3.5.

18 / 18

## 180bc17b86b108d6351327353ca789d0.pdf

MATHS 253, Lecture 2

2 March, 2022

Today’s topic: Basis, Dimension, The Coordinate Mapping

Colour code: Orange/Red

Lecture Notes: Sec. 1.4–1.5; pp. 11–16;

Textbook: §6.2, pp. 461–474.

1 / 19

Linear Independence

Definition 1
Let V be a vector space over a field F . A subset {v1, v2, . . . , vk} of V is said to be
linearly dependent if there exist scalars a1, a2, . . . , ak in F , not all of which are 0, such
that

a1v1 + a2v2 + · · · + ak vk = 0.

Otherwise it is linearly independent.

Example 2

• Let Eij be the matrix in Fm×n whose (ij)th entry is 1 and all other entries are 0.
Such a matrix is called a matrix unit. The set of all mn matrix units is linearly
independent.

• The set of monomials {1, x, x2, . . . , xn} is linearly independent in Fn[x].

2 / 19

Basis

Definition 3
Let V be a vector space. A basis B for V is a subset of V such that

• B is linearly independent set of vectors,
• B spans V .

Example 4

• The set of all mn matrix units Eij is linearly independent, spans Fm×n and hence a
basis of this vector space.

• The set of monomials {1, x, x2, . . . , xn} is linearly independent in Fn[x] and it also
spans Fn[x] so it is a basis of this vector space.

3 / 19

Exercise

Exercise 1
Do matrices

M1 =

[
1 0
0 1

]
, M2 =

[
1 0
0 −1

]
form a basis of the subspace D of all diagonal matrices from R2×2?
Yes, they do. They are linearly independent (one is not a multiple of another) and

a + b

2

[
1 0
0 1

]
+

a −b
2

[
1 0
0 −1

]
=

[
a 0
0 b

]
.

hence {M1, M2} spans D.

4 / 19

The Main Technical Result

Lemma 5
Suppose V = span{u1, . . . , um} and k > m. Then any k vectors v1, . . . , vk from V are
linearly dependent.

Proof. Let v1 = a11u1 + . . . + a1mum,

v2 = a21u1 + . . . + a2mum,

vk = ak1u1 + . . . + akmum.

Let us prove that there exist coefficients x1, . . . , xk , not all zero, such that
x1v1 + . . . + xk vk = 0. We write

x1v1 + . . . + xk vk =

x1(a11u1 + . . . + a1mum) + . . . + xk (ak1u1 + . . . + akmum) =

(a11x1 + . . . + ak1xk )u1 + . . . + (a1mx1 + . . . + akmxk )um = 0.

5 / 19

We notice that this equation will be satisfied if (x1, . . . , xk ) is a solution to the system
of linear equations

a11x1 + . . . + ak1xk = 0

. . .

a1mx1 + . . . + akmxk = 0

Since k > m, this system has more unknowns than equations, hence it has a nonzero
solution. This proves that {v1, . . . , vk} is linearly dependent and the lemma.

6 / 19

Exercise 2
Let V be a vector space. What is the smallest number k such that every k vectors of
V are linearly dependent in cases:

• V = R5 k = 6;
• V = R7[x] k = 9;
• V = R3×3 k = 10.

Theorem 6
All bases for a finite-dimensional vector space V are equinumerous (have the same
number of vectors).

Proof.
Suppose that B = {u1, . . . , um} and C = {v1, . . . , vk} be two bases for V and k > m.
Then By Lemma 5 C is linearly dependent, a contradiction.

7 / 19

Properties of Bases

Theorem 7
Let V be a finite-dimensional vector space. Then:

(a) Every spanning subset of V can be reduced to a basis;

(b) Every linearly independent subset of V can be extended to a basis.

Corollary 1

Every finite-dimensional vector space has a finite basis.

8 / 19

Dimension

Definition 8
Let V be a finite-dimensional vector space over F . The number of vectors in some
basis (which is the number of elements in all other bases too) is called the dimension
of V over F , denoted as dimF V or simply dim V .

Example 9

• dim Fn[x] = n + 1.
• dim Fm×n = mn.

9 / 19

Example
Let us consider the set U of all 2 × 2 matrices of the form[

a b
b a

]
, a, b ∈ R.

Then this is 2-dimensional subspace of R2×2. Indeed,[
a b
b a

]
= a

[
1 0
0 1

]
+ b

[
0 1
1 0

]
= aM1 + bM2.

Hence U is spanned by B = {M1, M2}. On the other hand B is linearly independent.
Suppose that

x1M1 + x2M2 = x1

[
1 0
0 1

]
+ x2

[
0 1
1 0

]
=

[
x1 x2
x2 x1

]
=

[
0 0
0 0

]
.

Then comparing entries of these matrices, we get x1 = x2 = 0. Hence B is indeed
linearly independent.

10 / 19

Coordinates relative to a basis

Theorem 10
Let B = {v1, v2, . . . , vn} be a basis of a vector space V over F . Then every vector u in
V can be uniquely represented as a linear combination of

u = a1v1 + a2v2 + · · · + anvn

with coefficients a1, a2, . . . , an from F . These coefficients are called the coordinates of
u relative to the basis B.

11 / 19

The coordinate mapping

Let B = {v1, v2, . . . , vn} be a basis of a vector space V over F , u be a vector in V and

u = a1v1 + a2v2 + · · · + anvn

Then we define the coordinate column of u relative to B as

[u]B =



a1
a2

an

 ∈ F n.

The mapping from V to F n

u 7→ [u]B
is called the coordinate mapping.

12 / 19

The Properties of the Coordinate Mapping

Theorem 11

1. The coordinate mapping is one-to-one and onto mapping from V to F n;

2. for any u, v from V and a scalar a in F

[u + v]B = [u]B + [v]B, [au]B = a[u]B.

Corollary 12

For any vectors v1, . . . , vk and for any scalars a1, . . . , ak

[a1v1 + . . . + ak vk ]B = a1[v1]B + a2[v2]B + . . . + ak [vk ]B.

13 / 19

The Main Property of the Coordinate Mapping

Theorem 13
Let V be a vector space over F . Then vectors u1, . . . , uk ∈ V are linearly dependent in
V if and only if their coordinate columns [u1]B, . . . , [uk ]B are linearly dependent in F

n.

First we observe that B = 0. Assume that the system {u1, . . . , uk} is linearly
dependent:

0 = a1u1 + . . . + ak uk

for certain coefficients a1, . . . , ak among which are not all zero. Applying the
coordinate mapping to both sides by Corollary 12 we obtain

0 = B = [a1u1 + . . . + ak uk ]B = a1[u1]B + a2[u2]B + . . . + ak [uk ]B,

i.e., the coordinate columns are also linearly dependent.

14 / 19

On the other hand, if we have a linear dependency among the coordinate columns

a1[u1]B + a2[u2]B + . . . + ak [uk ]B = 0,

again by Corollary 12 we may represent this equation as

[a1u1 + . . . + ak uk ]B = B

and deduce that a1u1 + . . . + ak uk = 0 because the coordinate mapping is one-to-one.

15 / 19

Exercise
Decide if the polynomials

f1(x) = 1 + x, f2(x) = x + x
2, f3(x) = 1 + x

2

form a linearly dependent or independent system.

Answer: Let us choose the standard basis B = {1, x, x2} of R2[x]. Then we form

([f1]B [f2]B [f3]B ) =

 1 0 11 1 0

0 1 1

 .

We have ∣∣∣∣∣∣
1 0 1
1 1 0
0 1 1

∣∣∣∣∣∣ = 2 6= 0.
This means {[f1]B [f2]B [f3]B} is linearly independent, hence {f1, f2, f3} is linearly
independent too.

16 / 19

Challenging Questions

1. Let V be a finite-dimensional subspace over R and W be a proper subspace of V .
Prove that W is also finite-dimensional.

2. Show that the vector space CR[a, b] of all continuous real-valued functions is not
finite dimensional.

17 / 19

To Summarise

As a result of today’s lecture (together with your further study) you should

• know the definition of linear dependence and linear independence;
• know the definition of a basis;
• understand the dimension;
• be able to find coordinates of a vector relative to a basis;
• know how to decide whether or not a vector is in the span of other vectors or be

able to decide if a system of vectors is linearly dependent or independent by using
the coordinate mapping.

• given a spanning subset of a subspace know how to find a basis – this will be
practiced in Tutorial 1.

18 / 19

What to do now

Consolidation: Try exercises: Lecture Notes: Exercises of Sec. 1.4–1.5; Textbook:
Exercises pp. 474-75: # 7,13 19, 29, 35, 45, 51.

Next topic: Linear Transformations

Colour code: Yellow

Preparation: Go through the text: Lecture Notes: Sec. 2.1–2.2. Textbook: §6.4, pp.
490–497; §6.6, pp. 515–521.

19 / 19

MATHS 253, Lecture 1

28 February, 2022

Today’s topic: Introduction. Fields, Vector Spaces and Subspaces
Colour code: Orange.
Lecture Notes: Sec. 1.1–1.3; pp. 7–11

Textbook: §6.1, pp. 447–459.

1 / 19

Goals of Algebra in 253
• Extend (slightly) the concept of scalar.
• Extend (significantly) the concept of vector space, most importantly

## Calculus homework help

Quiz 1
Math 263

Spring 2022

Instructions: Show work! You must convince me that you understand how to do the
problem and how the answer was obtained to get credit. In particular, answers (even
correct ones) unsupported by understandable work will receive no credit.

1. (10 points) If z = f (x,y), where x = r cosθ and y = r sinθ, find

(a) (3 points)
∂z

∂r

(b) (3 points)
∂z

∂θ

(c) (4 points)
∂2z

∂r∂θ

2. (10 points) Let f (x,y,z) = ex
2+2y2+3z2.

(a) (2 points) Let v⃗ =

3
13

,
4
13

,
12
13

. Find Dv⃗f (1,1,1).

(b) (3 points) In which direction is f increasing fastest at the point (1,1,1)?

(c) (2 points) What is the rate of increase of f in the direction of fastest increase?

(d) (3 points) Find the equation of the plane tangent to the ellipsoid

x2 + 2y2 + 3z2 = 6

at the point (1,1,1).

Page 2

3. (10 points) Suppose f : R 7→R is continuous and f (0) = 0. Consider the limit

lim
(x,y)→(0,0)

yf (x)

y2 + f (x)2
.

Does this exist for all such f , not exist for any such f , or exist for some such f ?
Explain.

Page 3

4. (10 points) Find the points on the surface

x2y2z = 1

closest to the origin.

Page 4

## Calculus homework help

Worksheet 1

Example Graphs in space may be represented in words, by equations, or visually. Consider the
equation

x2z − y2z − 16z − 3×2 − 3y2 = −48.

To find out what this graph looks like, let’s try to rewrite the equation:

x2z − y2z − 16z − 3×2 − 3y2 + 48 = 0
(z − 3)(x2 + y2 − 16) = 0

z − 3 = 0 or x2 + y2 − 16 = 0
z = 3 or z2 + y2 = 16

Thus we can describe this graph in words as the set of all points that either have z-coordinate 3
(these points form a plane) or are on the cylinder of radius 4 centered on the z-axis.

Write an equation for the graph described in words, and sketch the graph.

1. The sphere of radius 6 centered at (3, 0, −1)

2. The set of all points whose x and z coordinates are equal.

3. The cylinder of radius 2 centered on the line through the point (1, 2, 3) parallel to the y-axis.

After rewriting the equation as needed, describe the graph in words, then sketch its graph.

4. x2y − 4×2 − 5xy + 20x = 0

5. y2 − 2y + z2 + 6z = 15
Hint: Complete the square.

6. x2y + y3 + yz2 − 9y + 2×2 + 2y2 + 2z2 = 18

1