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


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]

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