Linear Algebra and Applications Homework # 2

ECE 269: Linear Algebra and Applications
Homework # 2


1. Suggested Readings. Following sections of Carl D. Meyer’s book “Matrix Analysis and
Applied Linear Algebra":
• Sections 4.5, 4.7: Properties of Rank, Linear Maps.
• Sections 3.5, 3.6, 3.7: Matrix multiplication, matrix inverse and properties.
Also review Summary Slides 1 and video lectures.
2. Problem 1: Convolution as Linear Map. Suppose that real-valued sequences {u(n)}
∞
n=−∞
and {v(n)}
∞
n=−∞ represent the input and output signals of a discrete-time linear timeinvariant system with impulse response h(n) ∈ R, n ∈ Z. Then, {u(n)} and {v(n)} are
related via convolution as
v(n) =
∞
∑
k=−∞
h(k)u(n − k), n ∈ Z.
Suppose that u(n) = 0 for n < 0 or n N, and define
x =













u(0)
u(1)
.
.
.
u(N)













and y =













v(0)
v(1)
.
.
.
v(N)













.
Thus x and y are vectors that capture N + 1 values of the input and output signals,
respectively.
∗For more information on Academic Integrity Policies at UCSD, please visit http://academicintegrity.ucsd.
edu/excel-integrity/define-cheating/index.html
(a) Find the matrix T such that
y = Tx
in terms of h(n).
(b) Describe the structure of T. Matrices of this structure are said to be Toeplitz.
3. Problem 2: Affine functions. A function f : Rn → Rm is said to be affine if for any
x, y ∈ Rn and any α, β ∈ R with α + β = 1, we have
f(αx + βy) = α f(x) + β f(y).
Note that without the restriction α + β = 1, this would be the definition of linearity.
(a) Suppose that A ∈ Rm×n and b ∈ Rm. Show that the function f(x) = Ax + b is affine.
(b) Prove the converse, namely, show that any affine function f can be represented
uniquely as f(x) = Ax + b for some A ∈ Rm×n and b ∈ Rm.
(Hint: Consider the linearity of the function g(x) = f(x) − f(0).)
4. Problem 3: Matrix multiplication. Let A, B ∈ Rn×n
. Prove or provide a counterexample
to each of the following statements.
(a) If AB = 0, then A = 0 or B = 0.
(b) If A2 = 0, then A = 0.
(c) If ATA = 0, then A = 0.
5. Problem 4: Linear Maps and Differentiation of polynomials. Let Pn be the vector space
consisting of all polynomials of degree ≤ n with real coefficient.
(a) Consider the transformation T : Pn → Pn defined by
T(p(x)) = dp(x)
dx .
For example, T(1 + 3x + x
2
) = 3 + 2x. Show that T is linear.
(b) Using {1, x, . . . , x
n} as a basis, represent the transformation in part (b) by a matrix
A ∈ R(n+1)×(n+1)
. Find the rank of A.
6. Problem 5: Rank of AAT
. Let A ∈ F
m×n
.
(a) Suppose that F = R. Prove that rank(AAT
) = rank(A) or provide a counterexample.
(b) Suppose that F = C. Repeat part (a).
(c) Suppose that F = C. Prove that rank(AAH) = rank(A) or provide a counterexample.
7. Problem 6: Left and Right Inverses.
(a) Show that if A ∈ Rm×n
is full-rank and tall, then ATA is nonsingular.
(b) Show that (ATA)
−1AT
is a left inverse of a full-rank tall matrix A ∈ Rm×n
.
(c) Let A ∈ Rm×n be full-rank and strictly tall (m n). Does it have a unique left
inverse? Prove or provide a counterexample.
(d) Show that if A ∈ Rm×n
is full-rank and fat, then AAT
is nonsingular.
(e) Show that AT
(AAT
)
−1
is a right inverse of a full-rank fat matrix A ∈ Rm×n
.
(f) Let A ∈ Rm×n be full rank and strictly fat (m < n). Does it have a unique right
inverse? Prove or provide a counterexample.