2020 — Assignment 8 (10 marks) — due 2 June

Question 1. [ 2 ]

Let v1 =

2

−1

0

, v2 =

−3

4

3

, x1 =

−4

−3

8

, x2 =

3

6

−5

.

Consider W = Span{v1, v2}. For each xi

:

(a) obtain a decomposition xi = xˇi + zi with xˇi = projW xi and zi ∈ W⊥ ; [1.5]

(b) compute the distance from xi to W . [0.5]

Question 2. [ 2 ]

Given vectors: a =

1

2

3

4

, z =

3

3

3

3

, b =

1

2

3

, y =

3

3

3

,

let A be a 3 × 4 matrix such that projW z = a where W = Col

ATA

(a) Find (and explain) a non-trivial solution for Ax = 0 [1]

(b) Explain whether or not projV y = b where V = Col

AAT

[1]

Question 3. [ 2 ]

Consider the following system of equations:

x1 + x3 = 1

2×2 + 2×3 = 1

−x1 + x3 = 1

(a) Find a least-squares solution to this system using normal equations [0.8]

(b) Obtain a QR factorisation of the matrix of this system [0.7]

(c) Use the QR factorisation to obtain a least-squares solution [0.3]

(d) Calculate the least-squares error of the solution [0.

Page 2

Question 4. [ 2 ]

Let

A =

0 2 0

1 0 3

(a) Find a singular value decomposition of A [1.8]

(b) Rewrite the decomposition as a linear combination of matrices with rank 1,

weighted with the singular values of A. [0.2]

Question 5. [ 2 ]

Consider the following quadratic form on R

3

:

Q(x) = 3x

2

1 + 3x

2

2 + 9x

2

3 − 8x2x3

(a) Write down the matrix of this quadratic form [0.1]

(b) Find the principal axes of this quadratic form [1.5]

(c) Write the change of variables transformation from x to y

that brings Q(x) to Q(y) with a diagonal matrix [0.2]

(d) Specify the diagonal matrix of Q(y) [0.2]