Description
1 Gaussian Elimination
Our aim is to solve the system of linear equations Ax = y. (General conditions for the existence of a
solution are given in FSP Appendix 2.B.1.) Comment on whether a solution to each of the following
systems of equations exists, and, if it does, find it.
(a)
A =
1 0 3
4 5 2
−1 −1 2
, y =
10
20
3
.
(b)
A =
1 0 2
4 5 8
−1 −1 −2
, y =
7
38
−9
.
(c)
A =
1 0 2
4 5 8
−1 −1 −2
, y =
1
2
3
.
2 Eigenvalues and Eigenvectors (Linear algebra refresher)
Let
A =
1 2
2 1
and B =
α β
β α
.
(a) Find the eigenvalues and unit-norm eigenvectors of A. Are the eigenvectors orthogonal? Check your
answer with a using Matlab.
(b) Compute the determinant of A, i.e. det A. Is A invertible? If it is, give its inverse; if not, say why.
(c) Find eigenvalues and unit-norm eigenvectors of B. For α ∈ {0, 1, 2, 3} and β ∈ [−3, 3], plot the
eigenvalues of B (using Matlab). (This will be four pairs of curves that are functions of one variable.)
(d) Compute the determinant of B. When is B invertible? For (α, β) ∈ [0, 5]2
, plot det B (with a
computer, using Matlab). (This will be a surface plot of a function of two variables.)
1 CIS 4930.006/CIS 6930.013 Homework 1
3 Multiplication by an orthogonal matrix
Consider the vector space R
n with standard norm and standard inner product. Prove that
(a) multiplication by an orthogonal matrix U preserves lengths, that is,
kUxk = kxk,
for any x.
(b) multiplication by an orthogonal matrix U preserves angles, that is,
hUx, Uyi = hx, yi,
for any x and y.
4 Bases and frames of R
2
Given the following sets of vectors:
Φ1 = {ϕ1,0, ϕ1,1} =
1
√
2
3
2
,
0
1
(1)
Φ2 = {ϕ2,0, ϕ2,1, ϕ2,2, ϕ2,3} =
(” 1
2
√
√
2
3
2
√
2
#
,
”
−
√
3
2
√
2
1
2
√
2
#
,
√
1
2
0
,
0
√
1
2
)
(2)
Φ3 = {ϕ3,0, ϕ3,1} =
1
√
2
3
2
,
−
√
3
2
1
2
(3)
Φ4 = {ϕ4,0, ϕ4,1, ϕ4,2} =
(
1
0
,
”
√
1
2
√
1
2
#
,
0
1
)
(4)
For each of the sets of vectors, Φ1 and Φ3, do the following:
(a) Write the matrix representation for the set, that is, the synthesis operator associated with the set.
(b) Find the dual basis. Sketch (in other words, draw the arrows that represent each vector of) the original
sets and their duals.
(c) Specify whether it is an orthonormal basis.
(d) For x = [2, 0]T, write down the projection coefficients, αi,k = hx, ϕei,ki.
(e) For the same x, verify the expansion formula ΦΦeT = I.
(f) Specify whether the expansion preserves the norm, that is, whether it is true that kxk =
P
k
|αi,k|
2
.
For each of the sets of vectors, Φ2 and Φ4, write the matrix representation for the set, that is, the synthesis
operator associated with the set.
5 Inner product
True or False, two vectors, say f(t) and g(t), are orthogonal if their inner product is zero. Using your
response to the above prove that f(t) = sin(πnt) and g(t) = sin(πmt) are orthogonal in the Hilbert space
L
2
[−1, +1], for any integers n 6= m (i.e., when n, m ∈ Z and n 6= m).
2
6 Inner product computation by expansion sequences
Let α and β be sequences in `
2 CIS 4930.006/CIS 6930.013 Homework 1
(N). Then, the functions
f(t) = α0 +
X∞
k=1
αk
√
2cos(2πkt),
g(t) = β0 +
X∞
k=1
βk
√
2cos(2πkt),
are in L
2
−
1
2
, +
1
2
. Demonstrate that the standard inner product between the functions, f(t) and
g(t) can be written as the standard inner product between the sequences α and β. That is, show that
hf(t), g(t)i = hα, βi.
Given the above results, write down (or derive, if you want) the norms of f(t) and g(t), i.e. write down
kf(t)k and kg(t)k.
7 Linear Independence (Optional, for extra credit.)
Find the values of the parameter a ∈ C such that the following set is linearly independent:
U =
0 a
2
0 j
,
0 1
1 a − 1
,
0 0
ja 1
.
For a = j, express the matrix
0 5
2 j − 2
as a linear combination of the elements of U. [Note that j denotes the imaginary unit, i.e. j = √
−1, so that
cj × dj = c × d × j
2 = c × d × −1 = −cd.]
8 Vector space C
n
(Optional, for extra credit.)
Prove that C
n is a vector space. Note: the symbol C means we are dealing with complex numbers. Thus,
a vector x ∈ C
n means that x is a vector with n entries and each of its entry is a complex number.
