Linear Algebra and Applications Homework # 3

ECE 269: Linear Algebra and Applications

Homework # 3


1. Suggested Readings. Review Summary Slides 2, video lectures and Discussion Problems.
2. Problem 1: Orthogonal Complement of a Subspace. Suppose that V is a subspace of Rn
.
Let
V
⊥ = {x ∈ R
n
: x
Ty = 0, ∀y ∈ V }
be the set of vectors orthogonal to every element in V.
(a) Verify that V
⊥ is a subspace of Rn
.
(b) Suppose that V = span(v1, v2, . . . , vk) for some v1, v2, . . . , vk ∈ Rn
. Express V and V
⊥
as subspaces induced by the matrix A =
?
v1 v2 · · · vk
?
∈ Rn×k and its transpose
AT
.
(c) Show that (V
⊥)
⊥ = V.
(d) Show that dim(V) + dim(V
⊥) = n.
(e) Show that V ⊆ W for another subspace W implies W⊥ ⊆ V⊥.
(f) Show that every x ∈ Rn
can be expressed uniquely as x = v + v
⊥, where v ∈ V and
v
⊥ ∈ V⊥. (Hint: Let v be the projection of x on V.)
3. Problem 2: Rank of a Product. Let A ∈ R4×3 has rank 2 and B ∈ R3×3 has rank 3.
(a) Find the smallest possible value rmin of rank(AB). Find specific A and B such that
rank(AB) = rmin.
(b) Find the largest possible value rmax of rank(AB). Find specific A and B such that
rank(AB) = rmax.
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4. Problem 3: An Inequality for Orthonormal Matrices: Suppose that the columns of U ∈
Rn×k are orthonormal (i.e. they are orthogonal and have unit l2 norm). Show that
kU
T
xk2 ≤ kxk2.
5. Problem 4: Householder Reflections. A Householder matrix is defined as
Q = I − 2uuT
for a unit vector u ∈ Rn
.
(a) Show that Q is orthogonal.
(b) Show that Qu = −u and that Qv = v for every v ⊥ u. Thus, the linear transformation y = Qx reflects x through the hyperplane with normal vector u.
(c) Given y, find x such that y = Qx.
(d) Show that det(Q) = −1.
Hint: Use the following properties of a determinant of a matrix: if AB is square (but
A and B do not have to be), then
det(I + AB) = det(I + BA).
(e) Given nonzero vectors x and y, find a unit vector u such that
Qx = (I − 2uuT
)x ∈ span(y),
in terms of x and y.
6. Problem 5: Projection Matrices. A symmetric matrix P = P
T ∈ Rn×n
is said to be a
projection matrix if P = P
2
.
(a) Show that if P is a projection matrix, then so is I − P.
(b) Suppose that the columns of U ∈ Rn×k are orthonormal. Show that UUT
is a projection matrix.
(c) Suppose that A ∈ Rn×k
is full-rank with k ≤ n. Show that A(ATA)
−1AT
is a projection matrix.
(d) Recall from lectures that the point y ∈ S ⊆ Rn
closest to x ∈ Rn
is said to be the
orthogonal projection (or projection in short) of x onto S. Show that if P is a projection
matrix, then y = Px is the projection of x onto R(P).
(e) Let u be a unit vector. Find the projection matrix P such that y = Px is the projection
of x onto span(u).