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SAINT MARY’S UNIVERSITY

DEPARTMENT OF MATHEMATICS AND COMPUTING

SCIENCE

MATH2321.2 Linear Algebra II

FINAL EXAMINATION

April 11 April 22, 2020

In Isolation Mode

Instructor: A. Finbow

Arthur Cayley

1821 – 1895

Sir William R. Hamilton

1805 1865

F. Georg Frobenius

1849 1917

Cayley in 1858 published Memoir on the theory of matrices in which, among other things, he proved that,

in the case of 2 × 2 matrices, a matrix satisfies its own characteristic equation. He stated that he had

checked the result for 3 × 3 matrices, indicating its proof, but said: I have not thought it necessary to

undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree. In

1853, Hamilton had already proven the 4 × 4 case in the course of his investigations into quaternions.

The general case was first proved by Frobenius (Schurs Doctoral supervisor) in 1878.

Instructions

1. Questions require a detailed solution. Sketchy solutions or mere answers will

not suffice. Include all steps so that the examiner has a clear indication of

how you arrived at each solution.

2. There are 221 possible marks.

3. The work that you submit must be your own: collaboration and/or plagiarism

is not permitted.

5. Due via e-mail: April 22, 2020 by 11:59pm.

HAPPY EASTER!!

1.

[1 point] Create your personal 4-dimensional vector f by placing the last 4 digits of your

A-number consecutively in the 4 entries of f as shown in this example.

?

EXAMPLE if youre A-number is A12345678 then your personal vector is(?).

?

?

2.

2

1

Let V be the subspace of R4 spanned by ? = {(1) , (0) , ?}, where f is your personal 41

2

1

0

dimensional vector from question 1.

a) [12 points] Use the Gram-Schmidt method to find an orthonormal basis for V.

b)

3.

4.

[8 points]

2

Use your result in part a) to find the projection of the vector (3) on V.

2

1

4

7

1

Consider the set of vectors S = {(2) , (5) , (8)}. As you all know, this is a dependent set

3

6

9

in the vector space R3.

a)

3

[8 points] Decide if S is independent set in ?11

(i.e. the 3-dimensional space in which

arithmetic is mode 11).

b)

3

[8 points] Decide if S is independent set in ?13

(i.e. the 3-dimensional space in which

arithmetic is mode 13).

a)

[8 points] Decide if the set S = {1 + x + x2 + x3, 1 – x + 2×2 – x3, 1 – 2x + x2 + 4×3} is

independent in P3.

b)

[8 points] Decide if 1 + 2x + 3×2 + 4×3 is in span(S).

c)

[8 points] Suppose that P3 is equipped with the inner product:

2

< ?, ? > = ?-1 ?(?)?(?)??. Find the projection of x + 2 onto x2.

5.

[10 points] Find the Error

Trial Theorem. Let W be a subspace of Rn. Then the projection matrix from Rn onto W is

the identity matrix.

Proof:

Let P be the projection matrix from Rn onto W. By Theorem 7.11, P = A (ATA)1 T

A . But then, using Theorem 3.9 c) we have

P = A (ATA)-1AT = A(A-1(AT) -1)AT = (AA-1)((AT) -1AT) = (I)(I) = I

QED

6.

1

[10 points] Let W be the subspace of R4 spanned by ? = {(-1) , ?}, where f is your

0

1

personal 4-dimensional vector from question 1. Find the 4 by 4 projection matrix P from Rn

onto W.

7.

a)

b)

8.

[10 points] Find the characteristic polynomial and the eigenvalue(s) for the matrix

1 -4

(

).

3 -2

8 -2 2

[15 points] Orthogonally diagonalize the following matrix B = (-2 5 4) [given

2

4 5

that its eigenvalues are ? and 9]. Your answer should consist of both an orthogonal

matrix Q and a diagonal matrix ? such that QTBQ = ?. Dont multiply out this

product!

Let T: M22? M22 be the linear transformation defined by

1 -2

T(?) = ? (

).

-3 6

a) [8 points] Find a basis for Ker(T).

b) [8 points] Find a basis for Im(T) (same as Range(T)).

c) [2 points] Show that T2(X) = 7T(X).

1 2

)

-3 4

d) [2 points] Use the result in c) [even if you did not get part c)], to find T 2020 (

0

1

Let B = {(

0

2 0

0

),(

),(

0

0 0

0

3

0 0

),(

)} be the basis for M22.

0

0 4

1 2

) with respect to the basis B.

-3 4

e) [2 points] Find the coordinates of (

f)

[8 points] Find the matrix representing T with respect to the basis B.

? ? ?

9. [9 points] Suppose that det( ? ? ? )= -2.

? ? ?

In each part, evaluate the expression.

y

z ?

? x

?

?

a) det ? 2r – x 2 s – y 2t – z ? =

? a

b

c ??

?

? ? ?

b) det( ? ? ? ) =

? ? ?

c)

10.

?

det(?

?

?

?

?

?

?) =

?

[10 points] Prove that if A and B are n by n matrices then det(AB) = det(BA).

11.

[2 points] Who was Schurs Doctoral supervisor?

12.

[28 points] This problem relates directly with the proof of Schurs Theorem. I am

assuming that you have the theorem in front of you and have read over the commentary

2

4

3

in the Notes for March 31 (part 1). I will take the matrix A = (-4 -6 -3) and tell

3

3

1

you that one of its eigenvalues is -2. You are to find a unitary (orthogonal in this case)

triangulation T for A following the method of the proof.

1. Compute a unit eigenvector u corresponding to -2.

2. Extend to an orthonormal basis for R3 with the first vector in the basis and form the

matrix U.

3. Multiply UTAU to find the 2 by 2 matrix B.

4. Find an eigenvalue for B and repeat the above steps to produce the matrix W

5. From this construct the matrix V, and obtain the product UV.

6. Finally write down the matrix T.

7. Now that you know all the eigenvalues, check if A is diagonalizable.

13. [10 points] Suppose that {v1, v2, v3} is an independent set in a vector space V. Prove that

{v1, v1 + 3v2, v1 + v2 + 2v3} is also independent.

SHORTER ANSWERS [26 Marks (2 for each part)]

In each part answer the question, evaluate the expression or indicate that there is not enough

information to proceed.

2-?

3? – 1

(

)

(

a) Compute the complex inner product where ? =

and ? =

-? )

5?

6? + 1

2

2-?

b) Normalize ( -? ).

6? + 1

c)

3 – 2?

?

Compute (

*

4

).

1 + 2?

d)

2 4

)

4 1

If A is a matrix with entries in Z5, compute det(A).

e)

If A is a matrix with entries in Z7, compute det(A).

f)

Let T: Rn?P2020 (the vector space of 3 by 3 matrices) be a linear transformation. If T is one

to one, what are the possible values for n?

g)

Let T: Rn? P2020 be a linear transformation. If T is onto, what are the possible values for n?

For the next 2 questions consider the matrix A = (

h) The Cayley-Hamilton Theorem states that a matrix satisfies its own characteristic equation.

Who was the person who proved the general case of this theorem and approximately how

old was he when the first proof of a special case was given?

i)

The 3 by 3 real symmetric matrix A has an eigenvalue ? = 2 of algebraic multiplicity one

and another eigenvalue ? of algebraic multiplicity two. A is also known to represent an

indefinite quadradic form and det(A) = 18. Find the value of ?.

For the next four questions suppose that A is a 3 by 3 matrix with eigenvalues 1 and 2.

Suppose also that the rank of A – I is equal to 1.

j) Which eigenvalue of A is repeated? EXPLAIN WHY

k) Write down a specific matrix that is similar to A and symmetric.

l) Write down a specific matrix that is similar to A and not symmetric. EXPLAIN WHY they

are similar.

m) Write down a specific matrix that has the same eigenvalues as A but is not similar to A.

EXPLAIN WHY they are not similar.

…

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