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linear algebra 2 final assignment
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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 (Schurs 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 youre 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 = ?. Dont 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 Schurs Doctoral supervisor?
12.
[28 points] This problem relates directly with the proof of Schurs 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.