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When applicable, box your final answer.
1 1
1- Find an SVD for ! 0 1%.
−1 1
1⁄3 2⁄3
2⁄3
⁄
⁄
2- Determine if !2 3 1 3 −2⁄3% is orthogonal. If it is, find the inverse.
2⁄3 −2⁄3 1⁄3
−3
1
3- Suppose 𝒚 = + - and 𝒖 = + -. Find the distance from 𝒚 to the line through 𝒖 and
9
2
the origin.
3
−4
1
−1
−2
1
4- Suppose 𝒚 = / 0, 𝒗𝟏 = / 0 and 𝒗𝟐 = / 0. Find the closest point to 𝒚 in the
1
−1
0
13
3
2
subspace 𝑊 spanned by 𝒗𝟏 and 𝒗𝟐 .
5- Find an invertible matrix 𝑃 and a matrix 𝐶 of the form +
has the form 𝐴 = 𝑃𝑉𝑃#$ .
𝑎
𝑏
−3 −1
−𝑏
- such that +
𝑎
2
5
4 0 2
6- Diagonalize !2 3 4% .
0 0 3
7- Let 𝑇 be a one-to-one linear transformation from a vector space 𝑉 into ℝ% . Show that
for 𝒖, 𝒗 in 𝑉, ⟨𝒖, 𝒗⟩ = 𝑇(𝒖) ⋅ 𝑇(𝒗) defines an inner product on 𝑉.
8- If 𝐴 is square and invertible, find a singular value decomposition of 𝐴#$ .
9- Show that the matrix 𝐴 and 𝐴& have the same characteristic polynomial.