? MATH 4435 & MATH 6435: LINEAR ALGEBRA II Yongwei Yao

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MATH 4435 & MATH 6435: LINEAR ALGEBRA II
HOMEWORK SETS AND EXAMS
Yongwei Yao
2014 FALL SEMESTER
GEORGIA STATE UNIVERSITY
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Contents
HW Set #01, Problems
HW Set #02, Problems
HW Set #03, Problems
HW Set #04, Problems
Midterm I, Review
Midterm I, Problems
HW Set #05, Problems
HW Set #06, Problems
HW Set #07, Problems
HW Set #08, Problems
Midterm II, Review
Midterm II, Problems
HW Set #09, Problems
HW Set #10, Problems
Extra Credit Set, Problems
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
There are four (4) problems in each homework set. Math 6435 students need to do all 4
problems while Math 4435 students need to do any three (3) problems out the four. If a
Math 4435 student submits all 4 problems, then one of the lowest score(s) is dropped. There
is a bonus point for Math 4435 students solving all 4 problems correctly/perfectly.
When solving homework problems, make sure that your arguments and computations are
rigorous, accurate, and complete. Present your step-by-step work in your solutions/proofs.
There is also a set of problems for extra credits; see the last page of this file.
There are three (3) PDF files for the homework sets and exams, one with the problems
only, one with hints, and one with solutions. Links are available below.
PROBLEMS
HINTS
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #01 (Due 09/03)
Problems
 


 
x1
1
2 −1 0
−3
x2 

6 −2 4 , b = −7 and x = 
Problem 1.1. Let A =  3
x3  . Then we construct
−2 −4 0 −8
5
x4
e
e
a new matrix A by A = A b , whose size is 3 × 5 (i.e., 3 by 5).
e
(1) Find the reduced row echelon form (RREF) of A.
(2) Find the reduced row echelon form (RREF) of A.
(3) Solve the equation Ax = b.
(4) Determine whether b ∈ Col(A), in which Col(A) stands for the column space of A.
 


x1
1
2 −1 0
x2 

6 −2 4  and x = 
Problem 1.2. Let A =  3
x3  as in Problem 1.1 above.
−2 −4 0 −8
x4
(1) Find rank(A), the rank of A.
(2) Find a basis for Col(A).
T
(3) Find the general solution of Ax = 0, in which 0 = 0 0 0 .
(4) Find a basis for Nul(A), in which Nul(A) stands for the null space of A.
 
 
 
 
7
5
3
1







Problem 1.3. Let b1 = 2 , b2 = 4 , b3 = 0 , and u = 8 in R3 . Then we construct
7
7
5
3
a 3 × 3 matrix A by A = b1 b2 b3 . Denote B = {b1 , b2 , b3 }.
(1) Compute det(A) (by cofactor expansion along the second row, for example).
(2) Determine whether B is a basis of R3 , i.e., whether b1 , b2 , b3 form a basis for R3 .
(3) Find [u]B , the coordinate vector of u with respect to B.
Problem 1.4 (Math 6435). Let V be a vector space (say over R), n a positive integer, and
B = {b1 , . . . , bn } ⊆ V such that both of the following conditions are satisfied:
(1) The vectors b1 , . . . , bn are linearly independent.
(2) For every v ∈ V , the vectors b1 , . . . , bn , v are linearly dependent.
Prove that B is a basis for V .
PROBLEMS
HINTS
1
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #02 (Due 09/10)
Problems
−2
4
8
6
Problem 2.1. Let b1 =
, b2 =
, c1 =
, and c2 =
in R2 . It is known that
1
3
11
7
2
both B = {b1 , b2 } and C = {c1 , c2 } are ordered
bases for R .
v
Let v ∈ R2 be a vector such that [v]C = 1 . Express [v]B in terms of v1 and v2 .
v2
Problem 2.2. Determine whether each of the following is a linear transformation (over R).
Show your reasoning.
x1
sin(x2 )
x
2
2
(1) L1 : R → R defined by L1
=
for all 1 ∈ R2 .
x
x1
x2
2 x1
|x1 |
(2) L2 : R2 → R2 defined by L2
=
, where |xi | is the absolute value of xi .
x2
|x2 |
(3) L3 : P2 → R defined by L3 (f (x)) = f (0) for all f (x) ∈ P2 .
(4) L4 : P2 → R defined by L4 (f (x)) = f (1) for all f (x) ∈ P2 .
Problem 2.3. Let V and W be vector spaces (say over R), v1 , . . . , vn ∈ V , and L a linear
transformation from V to W .
(1) Prove that if v1 , . . . , vn are linearly dependent, then so are L(v1 ), . . . , L(vn ).
(2) Prove that if L(v1 ), . . . , L(vn ) are linearly independent, then so are v1 , . . . , vn .
Problem 2.4 (Math 6435). Let V and W be vector spaces (say over R), v1 , . . . , vn ∈ V ,
and L a linear transformation from V to W such that Ker(L) = {0V }, in which 0V stands
for the zero vector of V .
Prove that if v1 , . . . , vn are linearly independent, then so are L(v1 ), . . . , L(vn ).
PROBLEMS
HINTS
2
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #03 (Due 09/17)
Problems
T
x1 − 2x2
Problem 3.1. Let L : R3 → R2 be defined by L(x) =
for all x = x1 x2 x3 .
2x2 − 3x3
Consider ordered bases B = {b1 , b2 , b3 } and C = {c1 , c2 } for R3 and R2 respectively, with
 
 
 
1
1
1
1
2






b1 = 0 , b2 = 1 , b3 = 1 ;
c1 =
, c2 =
.
2
3
0
0
1
Find C [L]B , the representation matrix of L with respect to ordered bases B and C.
R6
Problem 3.2. Define a linear transformation L : P3 → P2 by L(f (x)) = f 0 (x) + 0 f (x) dx
for all f (x) ∈ P3 . Consider ordered bases B = {1, x, x2 } and C = {1, x} for P3 and P2
respectively.
(1) Find C [L]B , the representation matrix of L with respect to ordered bases B and C.
(2) Express [L(a0 + a1 x + a2 x2 )]C in terms of a0 , a1 , a2 .
Problem 3.3. Let R2×2 be the vector space of all 2 × 2 matrices (over R). Consider
0 0
0 1
1 0
0 0
3 4
, E22 =
; and M =
.
, E21 =
, E12 =
E11 =
1 0
0 1
5 6
0 0
0 0
It is known (and easy to verify) that E = {E11 , E12 , E21 , E22} is an ordered
basis for R2×2
x
x
(over R). Define L : R2×2 → R2×2 by L(x) = M x for all x = 11 12 ∈ R2×2 .
x21 x22
(1) Find [M ]E , the coordinate vector of M with respect to the ordered basis E.
(2) Is L a linear transformation? There is no need to justify.
(3) Find the representation matrix of L with respect to E, if L a linear transformation.
Problem 3.4 (Math 6435). Let V and W be finite dimensional vector spaces (say over R)
with ordered bases B and C respectively. Let L : V → W be a linear transformation and A
be the representation matrix of L with respect to B and C, i.e., A = C [L]B .
Prove that, for every vector v ∈ V , v ∈ Ker(L) if and only if [v]B ∈ Nul(A).
PROBLEMS
HINTS
3
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #04 (Due 09/24)
Problems
x1 − 2x2
x
Problem 4.1. Let L : R2 → R2 be defined by L(x) =
for all x = 1 . Consider
3x1 − 4x2
x2
the standard ordered basis E = {e1 , e2 } and ordered basis B = {b1 , b2 } for R2 , with
1
0
1
2
e1 =
, e2 =
;
b1 =
, b2 =
.
0
1
2
3
(1) Find the matrix E [L]E representing L relative to the standard basis E.
P and P .
(2) Find E←B
B←E
(3) Find the matrix B [L]B representing L relative to B.
R6
Problem 4.2. Define a linear operator L : P3 → P3 by L(f (x)) = xf 0 (x) + 0 f (x) dx for
all f (x) ∈ P3 . Consider ordered bases E = {1, x, x2 } and B = {1, 1 + x, 1 + x + x2 } for P3 .
(1) Find the representation matrix E [L]E of L with respect to the standard basis E.
P and P .
(2) Find E←B
B←E
(3) Find the representation matrix B [L]B of L with respect to B.
Problem 4.3. Let R2×2 be the vector space of all 2 × 2 matrices (over R). Consider
0 0
0 0
0 1
1 0
;
, E22 =
, E21 =
, E12 =
E11 =
0 1
1 0
0 0
0 0
1 1
0 1
0 0
0 0
2 3
b1 =
,
b2 =
,
b3 =
,
b4 =
; and M =
.
1 1
1 1
1 1
0 1
4 5
It is known that E = {E11 , E12 , E21 , E22 } and B = {b1 , b2 , b3 , b4 } are ordered bases for
2×2
11 x12
R2×2 (over R). Define L : R2×2 → R2×2 by L(x) = xM for all x = [ xx21
.
x22 ] ∈ R
(1) Find the matrix E [L]E representing L relative to the standard basis E.
P and P .
(2) Find E←B
B←E
(3) Find the matrix B [L]B representing L relative to B.
Problem 4.4 (Math 6435). Let A and B be n × n matrices. Assume that A and B are
similar. Prove that A2 and B 2 are similar.
PROBLEMS
HINTS
4
SOLUTIONS
Math 4435/6435 (Fall 2014)
Midterm Exam I (10/01/2014)
Review
Fundamentals, basics: Problems 1.1, 1.2, 1.3, 1.4, 2.1. Topics include (reduced) row
echelon form, solving a system of linear equations, matrix, determinant, vector space,
subspace, rank of a matrix, dimension of a vector space, basis, coordinate vector, transition
matrix, etcetera.
Linear (in)dependence, basis: Problems 1.4, 2.3, 2.4.
Transition matrix: Problems 2.1, 4.1, 4.2, 4.3.
Linear transformation: Problems 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3.
Representation matrix: Problems 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3.
Similarity: Problems 4.1, 4.2, 4.3, 4.4.
Lecture notes and textbooks: All we have covered in class.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
as much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
PROBLEMS
HINTS
5
SOLUTIONS
Math 4435/6435 (Fall 2014)
Midterm Exam I (10/01/2014)
Problems
Problem I.1 (5 points). Consider the following vectors in R3 :
T
T
T
T
b1 = 1 2 3 , b2 = 4 0 6 , b3 = 7 8 9 ; and u = 1 0 3 .
It is known that B = {b1 , b2 , b3 } is an ordered basis for R3 .
(1) Compute det(A), where A = b1 b2 b3 .
(2) Find [u]B , the coordinate vector of u with respect to B.
Problem I.2 (5 points). Let R2×2 be the vector space of 2 × 2 matrices (over R). Consider
E11 = [ 10 00 ] , E12 = [ 00 10 ] , E21 = [ 01 00 ] , E22 = [ 00 01 ] ;
and M = [ 56 78 ] .
Denote E = {E11 , E12 , E21 , E22 }, which is an ordered basis for R2×2 (over R). Define a
linear transformation L : R2×2 → R2×2 by L(x) = xM for all x ∈ R2×2 .
(1) Find M E , the coordinate vector of M relative to the basis E.
[1 point]
(2) Find the representation matrix E [L]E of L with respect to basis E.
[4 points]
0 R 4
Problem I.3 (5 points). Define a linear operator L on P2 by L(f (x)) = xf (x) + 0 f (x) dx
for all f (x) ∈ P2 . Consider ordered bases E = {1, x} and B = {1 + 2x, 3 + 5x} for P2 .
(1) Find the representation matrix E [L]E of L with respect to the standard basis E.
P and P .
(2) Find E←B
B←E
(3) Find the representation matrix B [L]B of L with respect to the basis B.
Problem I.4 (5 points). Let V and W be vector spaces (say over R), L : V → W a linear
transformation, and v1 , . . . , vn , x ∈ V . Prove the following:
(1) If x ∈ Span{v1 , . . . , vn }, then L(x) ∈ Span{L(v1 ), . . . , L(vn )}.
(2) If L(x) ∈
/ Span{L(v1 ), . . . , L(vn )}, then x ∈
/ Span{v1 , . . . , vn }.
Extra Credit Problem I.5 (1 point, no partial credit). Let A and B be m × m matrices
over R, with m a positive integer. Prove that if A and B are similar, then A − λI and B − λI
are similar for every λ ∈ R.
PROBLEMS
HINTS
6
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #05 (Due 10/08)
Problems
Problem 5.1. Consider the following vectors in R3 (with the ordinary dot product):
T
T
T
x = −3 6 −5 ,
y = −3 −1 2 ,
z = 1 −2 −3 .
(1) Find x · y, the scalar product (dot product) of x and y.
(2) Find the angle between the two vectors y and z. Use radian as the unit.
(3) Which two vectors, if any, are orthogonal to each other?
Problem 5.2. Consider the following vectors in R3 (with the ordinary dot product):
T
T
x = −2 4 6 ,
y = −3 −1 2 .
(1) Find the vector projection of x onto y, denoted p.
(2) Find the scalar projection of x onto y, denoted α.
Problem 5.3. Consider the following vectors in R4 (with the ordinary dot product):
T
T
T
v1 = 1 2 −3 −9 ,
v2 = −2 −4 7 22 ,
v3 = −1 −2 1 1 .
Let V = Span{v1 , v2 , v3 }, the subspace spanned by v1 , v2 , v3 in R4 .
(1) Find a basis for V ⊥ , the orthogonal complement of V in R4 .
(2) Find dim(V ⊥ ).
Problem 5.4 (Math 6435). Let x, y ∈ Rn such that y 6= 0. Denote by p the vector
projection of x onto y.
(1) Prove that p ⊥ (x − p).
(2) Prove that kxk2 = kpk2 + kx − pk2 .
PROBLEMS
HINTS
7
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #06 (Due 10/15)
Problems
 
 
 
 
 
1
1
0
1
1
0
1
1
−2
2

 
 
 
 
Problem 6.1. Let v1 = 
1, v2 = 1, v3 = 0, b =  1 , and c = 3, which are
0
1
1
−2
4
4
vectors in R . Let A = v1 v2 v3 , which is a 4 × 3 matrix. Let V = Span{v1 , v2 , v3 }.
T
(1) Solve the equation Ax = b, with x = x1 x2 x3 ∈ R3 . Is b contained in V ?
T
(2) Solve the equation Ax = c, with x = x1 x2 x3 ∈ R3 . Is c contained in V ?
Problem 6.2. Let v1 , v2 , v3 , b, c, A and V be as in Problem 6.1.
(1) Find the least squares solutions of Ax = b.
(2) Find the least squares solutions of Ax = c.
Problem 6.3. Let v1 , v2 , v3 , b, c, A and V be as in Problem 6.1.
(1) Find min{kAx − bk : x ∈ R3 }, the minimum value of kAx − bk for all x ∈ R3 .
(2) Find min{kv − ck : v ∈ V }, the minimum value of kv − ck for all v ∈ V .
Problem 6.4 (Math 6435). Consider Rn , equipped with the ordinary inner product. Let
b ∈ Rn and let V be a vector subspace of Rn .
(1) Prove that there exists v0 ∈ V such that b − v0 ∈ V ⊥ .
(2) Given v0 as above, prove kb − v0 k 6 kb − vk for all v ∈ V .
PROBLEMS
HINTS
8
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #07 (Due 10/22)
Problems
 
 
 
 
1
1
0
8
0
1
1
7
4

 
 
 
Problem 7.1. Let v1 = 
1, v2 = 1, v3 = 0, and b = 6, which are vectors in R .
0
1
1
5
Let A = v1 v2 v3 , which is a 4 × 3 matrix. Let V = Span{v1 , v2 , v3 }.
(1) Find the general least squares solution of Ax = b, where x ∈ R3 .
(2) Find a vector v0 ∈ V such that kb − v0 k 6 kb − vk for all v ∈ V .
(3) Compute min{kb − vk : v ∈ V }.
R1
Problem 7.2. For all f (x), g(x) ∈ R[x], define hf (x), g(x)i = −1 f (x)g(x) dx. It is known
that this makes R[x] into an inner product space. We assume this inner product on R[x]
throughout Problem 7.2.
(1) Compute k1k.
(2) Compute kx2 k.
(3) Find cos θ, where θ is the angle between 1 and x2 .
(4) Find the vector projection p of 1 onto x2 .
Problem 7.3. Let V be an inner product space (hence ha, bi is defined for all a, b ∈ V ).
Let v1 , . . . , vn , x ∈ V and S = Span{v1 , . . . , vn }.
Prove that x ∈ S ⊥ if and only if x ⊥ vi for all i = 1, . . . , n.
Problem 7.4 (Math 6435). Let W be an inner product space (hence hx, yi and kxk are
defined for all x, y ∈ W ), V be a subspace of W and b ∈ W . Assume W = V ⊕ V ⊥ .
Prove that there exists p ∈ V such that kb − pk 6 kb − zk for all z ∈ V .
PROBLEMS
HINTS
9
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #08 (Due 10/29)
Problems
Problem 8.1. Let {u1 , u2 , u3 } be an orthonormal basis for an inner product space V .
Consider the following vectors in V :
x = −u1 − u2 + 2u3 ,
y = 2u1 − 2u3 ,
z = c1 u1 + c2 u2 + c3 u3 .
(1) Find the angle θ between x and y. Use radian as the unit.
(2) Assume z ⊥ u1 , hz, u2 i = −3 and kzk = 5,. Find the possible values of c1 , c2 and c3 .


1 3 4
1 3 2 

Problem 8.2. Apply the Gram-Schmidt process to the columns of A = 
1 1 0  in
1 1 −2
order to find an orthonormal basis for Col(A). (Assume the usual inner product on R4 .)
R1
Problem 8.3. Consider the inner product on P3 with hf (x), g(x)i = −1 f (x)g(x) dx for all
f (x), g(x) ∈ P3 . (We assume this inner product on P3 throughout Problem 8.3.)
Apply the Gram-Schmidt process to {1, x, x2 } to find an orthonormal basis for P3 .
Problem 8.4 (Math 6435). Let V be an inner product space and S a subspace of V . Assume
that S has an orthonormal basis {u1 , . . . , un }. Let x ∈ V be an (arbitrarily) given vector.
Prove that there exist y ∈ S and z ∈ S ⊥ such that x = y + z.
PROBLEMS
HINTS
10
SOLUTIONS
Math 4435/6435 (Fall 2014)
Midterm Exam II (11/05/2014)
Review
Materials covered earlier: Homework Sets 1, 2, 3, 4; Midterm I.
Scalar/dot product, norm, projection, orthogonality: Problems 5.1, 5.2, 5.3, 5.4.
Least squares solutions: Problems 6.1, 6.2, 6.3, 6.4, 7.1.
Inner product, norm, projection, orthogonality: Problems 7.2, 7.3, 7.4.
Orthonormal bases, Gram-Schmidt process: Problems 8.1, 8.2, 8.3.
Proofs involving orthogonality: Problems 5.4, 6.4, 7.3, 7.4, 8.4.
Lecture notes and textbooks: All we have covered.
Note: The above list is not intended to be complete. The problems in
the actual test may vary in difficulty as well as in content. Going over,
understanding, and digesting the problems listed above will definitely help.
However, simply memorizing the solutions of the problems may not help you
as much.
You are strongly encouraged to practice more problems (than the ones
listed above) on your own.
PROBLEMS
HINTS
11
SOLUTIONS
Math 4435/6435 (Fall 2014)
Midterm Exam II (11/05/2014)
Problems
 
 
 
 
1
0
0
0
0
1
0
−5
4

 
 
 
Problem II.1 (5 points). Consider v1 = 
0, v2 = 0, v3 = 1, and b =  3  in R .
1
1
1
6
Let A = v1 v2 v3 , which is a 4 × 3 matrix. Let V = Span{v1 , v2 , v3 }.
T
(1) Find the general least squares solution of Ax = b, with x = x1 x2 x3 ∈ R3 .
(2) Calculate min{kb − zk : z ∈ V }, the minimum value of kb − zk for all z ∈ V .
R2
Problem II.2 (5 points). For all f (x), g(x) ∈ R[x], define hf (x), g(x)i = 0 f (x)g(x) dx,
making R[x] an inner product space. Use this inner product throughout Problem II.2.
(1) Compute kxk.
(2) Compute kx2 k.
(3) Calculate cos θ, where θ is the angle between x and x2 .
(4) Find the vector projection p of x2 onto x.
Problem II.3 (5 points). Apply the Gram-Schmidt process to a1 , a2 and a3 in R3 , where
 
 
 
−7
6
3





and a3 = −1 .
a2 = 8
a1 = 4 ,
2
9
0
(Throughout this problem, we assume the inner product to be the dot product on R3 .)
Problem II.4 (5 points). Let V be an inner product space over R (so hu, vi is defined for
all u, v ∈ V ) and let a, b ∈ V \ {0}. Denote by θ the angle between a and b.
Prove the following equation: ka + bk2 = kak2 + 2 kak kbk cos θ + kbk2 .
Extra Credit
Problem
II.5
 credit). Consider the following vectors
 (1 point, nopartial
 

3
1
0
1
5
1
1
0
 
 
 
 
, v2 = 1, v3 = 1, and u =  4  in R5 . Let V = Span{v1 , v2 , v3 } ⊆ R5 .
1
v1 = 
 
 
 
 
0
−7
1
1
11
1
0
1
⊥
Find vectors v ∈ V and w ∈ V such that u = v + w. (Assume the dot product on R5 .)
PROBLEMS
HINTS
12
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #09 (Due 11/12)
Problems


1
0
3
−1 3 −6
. Find the QR factorization of A; that is, find a
Problem 9.1. Let A = 
−1 0
1
1 −3 2
4 × 3 matrix Q whose columns are orthonormal and an upper triangular 3 × 3 matrix R with
positive diagonal entries such that A = QR.
−6 −2
Problem 9.2. Let A =
. Complete the following:
9
5
(1) Find the characteristic polynomial of A.
(2) Find all the eigenvalues of A.
(3) For each eigenvalue of A, find the corresponding eigenspace (of A).


2
3
7
Problem 9.3. Let B =  0 −4 0 . Complete the following:
−5 1 −8
(1) Find the characteristic polynomial of B.
(2) Find all the real eigenvalues of B.
(3) For each real eigenvalue of B, find a corresponding eigenvector (of B).
Problem 9.4 (Math 6435). Let A ∈ Cn×n be an n × n matrix and λ be an eigenvalue of A.
(1) Prove that λ2 is an eigenvalue of A2 .
(2) Prove that if A2 = 0 then λ = 0.
PROBLEMS
HINTS
13
SOLUTIONS
Math 4435/6435 (Fall 2014)
Homework Set #10 (Due 11/19)
Problems
1 −2
Problem 10.1. Let C =
∈ C2×2 . Complete the following:
10 5
(1) Find the characteristic polynomial of C.
(2) Find all the (complex) eigenvalues of C.
(3) For each (complex) eigenvalue of C, find the corresponding eigenspace (of C).
−6 −2
1 −2
Problem 10.2. Let A =
and C =
as in Problem 9.2 and Problem 10.1.
9
5
10 5
(1) Determine whether A is diagonalizable over C. If so, find a non-singular matrix X
and a diagonal matrix D such that X −1 AX = D.
(2) Determine whether A is diagonalizable over C. If so, find a non-singular matrix Y
and a diagonal matrix E such that Y −1 CY = E.


2 − 3i
0
0
2 − 3i 0 ∈ C3×3 . Complete the following:
Problem 10.3. Let J =  1
0
0
4
(1) Find all the (complex) eigenvalues of J.
(2) For each (complex) eigenvalue of J, find the corresponding eigenspace (of J).
(3) Is J diagonalizable? If so, find an invertible matrix X such that X −1 JX is diagonal.
Problem 10.4 (Math 6435). Let A = aij n×n ∈ Cn×n and assume that A is Hermitian.
(1) Prove that the diagonal entries of A (i.e., aii for 1 6 i 6 n) are real numbers.
(2) Prove that, for every B ∈ Cn×m , B H AB is a Hermitian matrix of size m × m.
PROBLEMS
HINTS
14
SOLUTIONS
Math 4435/6435 (Fall 2014)
Extra Credit Set
Problems
You must solve a problem completely and correctly in order to get the extra credit. You
may attempt/submit a problem for as many times as you wish by December 03, 2014.
The points you get here will be added to the total score from the homework assignments.
Problem E-1 (3 points). Let V and W be vector spaces, L : V → W a linear transformation,
u1 , . . . , um ∈ Ker(L), and v1 , . . . , vn ∈ V . Assume that u1 , . . . , um are linearly independent
and that L(v1 ), . . . , L(vn ) are linearly independent. Prove that u1 , . . . , um , v1 , . . . , vn are
linearly independent.
Problem E-2 (3 points). Are [ 20 12 ] and [ 10 24 ] similar?
Problem E-3 (3 points). Prove or disprove: For all n × n matrices A, B and C, if A and
B are similar then AC and BC are similar.
Problem E-4 (3 points). Let V be the set of all real functions defined on [−π, π] whose
n-th derivativesR exist over [−π, π] for all n > 0. Note that V is a vector space over R. Define
π
hf (x), g(x)i = −π f (x)g(x) dx for all f (x), g(x) ∈ V . This makes V into an inner product
space. Let S = Span{cos(nx) | n ∈ N}. Prove or disprove: V = S ⊕ S ⊥ .
Problem E-5 (3 points). Consider Cn , an inner product space over C with hx, yi = yH x.
Let S be a subspace of Cn and A ∈ Cn×n . Assume that A is Hermitian and S is invariant
under A (meaning Ax ∈ S for all x ∈ S). Prove that S ⊥ is invariant under A (meaning
Ax ∈ S ⊥ for all x ∈ S ⊥ ).
Problem E-6 (3 points). Let A ∈ Cn×n and assume that A is normal (i.e., AAH = AH A).
Prove or disprove: For every eigenvalue λ of A, the corresponding eigenspace V (of A) is
invariant under AH .
PROBLEMS
HINTS
SOLUTIONS