MAT2611 Assignment 4 Solutions 2026

MAT2611 Assignment 4 Solutions 2026
UNISA
DUE DATE: 27 JULY 2026

,Problem 1
The inner product on ℝ2 is generated by the matrix
0 1
𝐴=[ ]
1 2

where
1 1 3
𝐮 = [ ] , 𝐯 = [ ] , 𝐰 = [ ].
3 2 1

For an inner product generated by a matrix,
⟨𝑥, 𝑦⟩ = 𝑥 𝑇 𝐴𝑦.



(a) Calculate ⟨𝒗, 𝒖⟩
First calculate 𝐴𝑢:
0 1 1 (0)(1) + (1)(3) 3
𝐴𝑢 = [ ][ ] = [ ]=[ ]
1 2 3 (1)(1) + (2)(3) 7

Now compute
3
𝑣 𝑇 (𝐴𝑢) = [1 2] [ ]
7
= (1)(3) + (2)(7)
= 3 + 14
= 17

Answer
⟨𝑣, 𝑢⟩ = 17



(b) Calculate ⟨𝒘, 𝒗 + 𝟐𝒖⟩
First find
1 2
2𝑢 = 2 [ ] = [ ]
3 6

Hence,
1 2 3
𝑣 + 2𝑢 = [ ] + [ ] = [ ]
2 6 8

Now calculate
0 1 3 8
𝐴(𝑣 + 2𝑢) = [ ][ ] = [ ]
1 2 8 19

Therefore,

, ⟨𝑤, 𝑣 + 2𝑢⟩ = 𝑤 𝑇 𝐴(𝑣 + 2𝑢)
8
= [3 1] [ ]
19
= (3)(8) + (1)(19)
= 24 + 19
= 43

Answer
⟨𝑤, 𝑣 + 2𝑢⟩ = 43



(c) Calculate ∥ 𝒘 ∥𝟐
By definition,
∥ 𝑤 ∥2 = ⟨𝑤, 𝑤⟩

First calculate
0 1 3 1
𝐴𝑤 = [ ][ ] = [ ]
1 2 1 5

Hence,
1
𝑤 𝑇 𝐴𝑤 = [3 1] [ ]
5
= (3)(1) + (1)(5)
=8

Therefore,
∥ 𝑤 ∥2 = 8



(d) Calculate 𝒅(𝒗, 𝒖)
The distance is
𝑑(𝑣, 𝑢) =∥ 𝑣 − 𝑢 ∥

First calculate
1 1 0
𝑣−𝑢 =[ ]−[ ]=[ ]
2 3 −1

Now calculate
0 1 0 −1
𝐴(𝑣 − 𝑢) = [ ][ ] = [ ]
1 2 −1 −2

Then