18.06 Spring2019 short pset 3

due Wednesday 2/27 at 10:55am

This problem set is completely based on lectures and your ingenuity. No foreshadowing. You can optionally not use Juliabox at all, though using Julia may enhance the experience. This entire set can be done with pencil and paper if you prefer. There should be no mention of AAᵀ,AᵀA or eigenvalues. You should not google or use wikipedia, just what you learned in lecture.

1. Consider the matrix A defined below.

A = [1 4 2;2 8 4;-1 -4 -2]

3×3 Array{Int64,2}:
  1   4   2
  2   8   4
 -1  -4  -2

You should be able to describe the column space of A without any fancy svd's, just by common sense.
1a. Please describe the column space.

This matrix can be expressed as an outer product, A=xyᵀ.
1b. Find an x and y such that A=xyᵀ.

Without a computer you should be able to write down a compact (rank r format) svd of A.
1c. What is U, Σ, and V? Write an exact answer not a decimal.

If you wish to check your work, or are not sure what an exact answer would look like, one option (nothing required here!) is to execute the Julia code, but you will get decimals which you will have to think a bit about to recognize as exact numbers.

using LinearAlgebra
r = rank(A)
U,s,V = svd(A)
display(U[:,1:r])
display(s[1:r])
display(V[:,1:r])

2. Suppose we have the matrices U,Σ, and V as follows:

(Note dividing a matrix by √10 means divide every entry by the square root of 10). Suppose A=UΣVᵀ.

U = [1 -3;3 1] / √10
Σ = [√45 0; 0 √5]
V = [1 -1;1 1] / √2

2×2 Array{Float64,2}:
 0.707107  -0.707107
 0.707107   0.707107

2a. Is A invertible? (Do not multiply through) Why or why not?

2b. What is the column space of A?

2c. What is the nullspace of A?

2d. In factored form (no need to multiply out) write the rank 1 approximation to A corresponding to the image compression notebook we saw in class.

2e. Find an SVD of the matrix AV without multiplying out AV

2f. Find an SVD of the matrix A*V*V (you can multiply out V*V using pencil and paper)

You can optionally check your work using Julia.

3. Let A be an arbitrary mxn matrix.

3a. Are the singular values of 2A always double that of A? If yes, why? If not give a counterexample.

3b. Are the singular values of -2A always -2 those of A? If yes, why? If not give a counterexample. (Be careful.)

4a. Let A=[1 2 3].

Describe the nullspace of A precisely as which geometric object with what normal?

4b. Let A=[1 2 3]ᵀ. What is the nullspace of A?