## 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

## 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).

## 6 Inner product computation by expansion sequences

Let α and β be sequences in `

2

(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.