Linear Algebra and Applications Homework # 1

ECE 269: Linear Algebra and Applications
Homework # 1

Collaboration Policy: This homework set allows limited collaboration. You are expected
to try to solve the problems on your own. You may discuss a problem with other students to
clarify any doubts, but you must fully understand the solution that you turn in and write it up
entirely on your own. Blindly copying another student’s result will be considered a violation
of academic integrity. ⇤
1. Suggested Reading. Sections 4.1 4.4 of Carl D. Meyer’s book “Matrix Analysis and
Applied Linear Algebra".
2. Problem 1: Vectors Spaces other than RN.
Show that the following sets with operations defined over the given fields are not valid
vector fields.
(a) The set of rational numbers defined over R.
(b) The set of polynomials {a0 + a1x + a2x2 | a0, a1, a2 2 R+} defined over R, where R+
is the set of positive real numbers
(c) The set of vectors in R2 defined over R with vector addition and scalar multiplication
defined as follows. Here a, b, c, d,r 2 R
i.
2
6
6
4
a
b
3
7
7
5
+
2
6
6
4
c
d
3
7
7
5 =
2
6
6
4
a + c
b + d
3
7
7
5
and r.
2
6
6
4
a
b
3
7
7
5 =
2
6
6
4
r.a
b
3
7
7
5
ii.
2
6
6
4
a
b
3
7
7
5
+
2
6
6
4
c
d
3
7
7
5 =
2
6
6
4
a + c
b + d
3
7
7
5
and r.
2
6
6
4
a
b
3
7
7
5 =
2
6
6
4
r.a
0
3
7
7
5
iii.
2
6
6
4
a
b
3
7
7
5
+
2
6
6
4
c
d
3
7
7
5 =
2
6
6
4
0
0
3
7
7
5
and r.
2
6
6
4
a
b
3
7
7
5 =
2
6
6
4
r.a
r.b
3
7
7
5
iv.
2
6
6
4
a
b
3
7
7
5
+
2
6
6
4
c
d
3
7
7
5 =
2
6
6
4
a c
b d
3
7
7
5
and r.
2
6
6
4
a
b
3
7
7
5 =
2
6
6
4
r.a
r.b
3
7
7
5
⇤For more information on Academic Integrity Policies at UCSD, please visit http://academicintegrity.ucsd.
edu/excel-integrity/define-cheating/index.html
3. Problem 2: Adjacency graph.
Consider a simple graph (an undirected graph with no self loops or multiple edges) with
n vertices. Let A 2 Rn⇥n be the adjacency graph, defined by
Aij =
(
1 if there is an edge between vertex i and vertex j,
0 otherwise.
Note that A = AT and Aii = 0, i = 1, 2, . . . , n, since there are no self loops. Let B = Ak,
where k 2 Z. Give a simple interpretation of Bij in terms of the original graph and k, and
justify your answer. (Hint: Use the concept of a path between two nodes.)
4. Problem 3: Vector Spaces of Polynomials.
Consider the set Pn(R) of all real valued polynomials of degree  n with real coefficients:
Pn(R) = { f(x) =
n
Â
k=0
ckxk
, c0, c1, ··· , cn 2 R} (1)
(a) Show that Pn(R) is a vector space. What is the dimension of this vector space?
(b) Is the union [m
n=1Pn a vector space? Does this contradict or comply with something
you learned in class?
(c) Find a basis for P4 containing {x2 + 1, x2 1}
(d) Find a basis for P2 from the set {1 + x, x + x2, x + 2x2, 2x + 3x2, 1 + 2x + x2}
5. Problem 4: Symmetric and Hermitian matrices.
A square matrix A is said to be symmetric if its transpose AT satisfies AT = A, and a
complex-valued square matrix A is said to be Hermitian if its conjugate transpose AH =
(A)T = AT satisfies AH = A. Thus, a real-valued square matrix A is symmetric if and
only if it is Hermitian. Which of the following is a vector space?
(a) The set of all n ⇥ n real-valued symmetric matrices over R.
(b) The set of all n ⇥ n complex-valued symmetric matrices over C.
(c) The set of all n ⇥ n complex-valued Hermitian matrices over R.
(d) The set of all n ⇥ n complex-valued Hermitian matrices over C.
For each case, either verify that it is a vector space or prove otherwise.
6. Problem 5: Properties of Vector Spaces.
(a) Prove that the additive inverse of an element in a vector space is unique.
(b) Prove that adding a vector v to a set of vectors S={w1, w2, ..., wN} will not add new
vectors to span(S), if and only if v 2 span(S).
7. Problem 6: Linear Independence.
(a) Consider the stacked vectors
z1 =
2
6
6
4
x1
y1
3
7
7
5 , ··· , zn =
2
6
6
4
xn
yn
3
7
7
5 ,
i. Suppose x1,..., xn are linearly independent (no assumption is made on y1,..., yn).
Can we conclude that the vectors z1,..., zn are linearly independent? If yes, provide a proof. If no, give a counterexample.
ii. Suppose x1,..., xn are linearly dependent (no assumption is made on y1,..., yn).
Can we conclude that the vectors z1,..., zn are linearly dependent? If yes, prove
the result. If no, give a counterexample.
8. Problem 7: Finding Basis.
Find a basis for each of the following subspaces of Rn.
(a) Subspace S, which is the intersection of U and V, where U = span{(2, 1, 3, 0)T,(1, 0, 1, 0)T}
and V = span{(0, 1, 0, 0)T,(0, 0, 1, 0)T,(0, 0, 0, 1)T}.
(b) All vectors whose components are equal.
(c) All vectors whose components sum to zero.
(d) All vectors spanned by 
1100T
,

0110T
,

0011T
, and 
1001T