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MAT2611 Assignment 4 Solutions 2026

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Course Linear Algebra 2 (MAT2611)
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MAT2611 Assignment 4 Solutions 2026 0-7-9-3-2-2-6-4-2-7 unisa due date: 27 july 2026

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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

Table of contents

  1. 01 Problem 1: Inner Product on ℝ² 2
    1. Part (a): Calculate ⟨v, u⟩ 2
    2. Part (b): Calculate ⟨w, v + 2u⟩ 2
    3. Part (c): Calculate ∥w∥² 3
    4. Part (d): Calculate d(v, u) 3
    5. Part (e): Calculate ∥w − 2v∥ 4
  2. 02 Problem 2: Inner Product on P₂ with Integration 5
    1. Part (a): Calculate d(u, v) 5
    2. Find u − v 5
    3. Form the inner product 5
    4. Expand and integrate 6
    5. Evaluate the definite integral 7
    6. Part (b): Calculate ∥v∥² 7
    7. Expand the polynomial 8
    8. Integrate and evaluate 8
  3. 03 Problem 3: Mutual Orthogonality in ℝ³ 9
    1. Compute u · v 9
    2. Compute v · w 9
    3. Verify u · w = 0 10
    4. Conclusion 10
  4. 04 Problem 4: Gram–Schmidt Process on P₂ 11
    1. First Orthogonal Vector 11
    2. Compute ⟨u₁, u₁⟩ 11
    3. Second Orthogonal Vector 11
    4. Calculate ⟨v₂, u₁⟩ 12
    5. Compute projection coefficient 12
    6. Verify orthogonality u₂ ⊥ u₁ 13
    7. Compute ⟨u₂, u₂⟩ 13
    8. Third Orthogonal Vector 14
    9. First projection: Compute ⟨v₃, u₁⟩ 14
    10. Second projection: Compute ⟨v₃, u₂⟩ 15
    11. Compute u₃ 16
    12. Verify orthogonality of u₃ 16
    13. Final Orthogonal Basis 17
  5. 05 Problem 5: Orthogonal Matrix Construction 18
    1. Observations on Matrix Structure 18
    2. Compute R₂ · R₃ 18
    3. Analysis of Orthogonality Condition 19
    4. Conclusion and Uniqueness 19
  6. 06 Problem 6: Spectral Decomposition 20
    1. Find the Eigenvalues 20
    2. Compute characteristic polynomial 20
    3. Solve the quadratic equation 20
    4. Find Eigenvector for λ₁ 21
    5. Normalise v₁ 21
    6. Find Eigenvector for λ₂ 22
    7. Normalise v₂ 23
    8. Form the Orthogonal Matrix P 23
    9. Form the Diagonal Matrix D 23
    10. Spectral Decomposition Formula 23
    11. Alternative Spectral Decomposition 23

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