Math 318 Homework 7

发布时间：2024-05-25

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Math 318 Homework 7

(1) Differentiation is a linear transformation

We showed in class that the set of all univariate polynomials of degree at most d, R[x]≤d is a vector space. Consider a general polynomial of degree d:

f(x) = ad xd + ad一1 xd一1 + … + a1 x + a0 .

Recall that we can identify f(x) with its vector of coefficients

f = (a0 , a1 , a2 ,..., ad ) E Rd+1 .

(a) Compute the derivative f\ (x) and write down its vector of coefficients.

(b) Argue that the function Dd : R[x]≤d → R[x]≤d一1 that sends f(x) ~ f\ (x) is a linear transformation.

(d) Using the previous calculation write down the matrix Md of the linear transformation Dd.

(e) What does the M5 matrix look like?

(f) Using M5 express the derivative of 5x5 一 19x3 + 24x 一 3 as an image of the linear transformation D5 .

(2) SVD of Symmetric and PSD matrices

(a) Compute the SVD of the symmetric matrix (using Julia or otherwise)

B = 3(2) 5(4) 6(5) .

(b) If A is a symmetric matrix of size nx n, argue that σi = ∣λi ∣ for all i. Here σi is the ith singular value of A and λi is the ith eigenvalue of A.

general symmetric matrix C to the SVD of C? Say in words what steps need to be taken.

(d) If A is a PSD matrix of size nx n then what is the relationship between its singular values and eigenvalues? What is the SVD of A?

(3) Rank one matrices

(a) Argue that for any two matrices A and B, rank(A + B) ≤ rank(A) + rank(B). Hint: What can you say about the columns of A + B and the column space of A + B in relation to the column spaces of A and B? How does the dimension of Col(A + B) relate to the sum of the dimensions of Col(A) and Col(B). You

can use the fact that if S and T are two sets of vectors in Rn then dim(span{S nT}) ≤ dim(span{S}) + dim(span{T}).

(b) Use SVD to argue that every rank one matrix in Rmxn is of the form uvL for u E Rm and v E Rn.

(d) If the columns of A E Rmxk are a1 , . . . , ak and the rows of B E Rk xn are b1(L) , . . . , bk(L) argue that AB = a1 b1(L) + a2 b2(L) + … + akbk(L) .

Hint: You could show that the (i,j)-entry on the left side is the same at the (i,j)-entry on the right side. Warm up by checking that if

A = [c(a) d(b)] and B = [h(e) i(f) j(g)]

then

AB = [c(a)] [e f g] + [d(b)] [h i j] .

(4) t Projection with an orthonormal basis

In class we learned that if V ⊆ Rn is a subspace with basis a1 , . . . , ak and A E Rn ×k is the matrix with columns a1 , . . . , ak , then projection onto V is achieved by the

linear transformation with matrix A(A⊺ A)−1A⊺ . In this exercise we are going to see how this formula simplifies if we had started with an orthonormal basis of V.

(a) Suppose q1 , . . . , qk is an orthonormal basis of V and Q E Rn ×k is the matrix with columns q1 , . . . , qk .

(i) Show that the projection matrix P = Q(Q⊺ Q)−1Q⊺ is

q1 q1(⊺) + q2 q2(⊺) + … + qk qk(⊺) .

(ii) Using (i) compute projVb, the projection of b E Rn onto V. (Your answer should be a linear combination of q1 , . . . , qk .)

(iii) From (ii), what are the coordinates of projVb in the basis q1 , . . . , qk?

(iv) Use your knowledge of orthogonal projectors to write down the matrix that projects onto V⊥ .

(v) Using this projector to find projV⊥ b.

(b) Suppose we find additional vectors so that q1 , . . . , qk , qk+1 , . . . , qn is an orthonormal basis of Rn. Check for yourself that {qk+1 , . . . , qn } is an orthonormal basis of V⊥ .

(i) Apply what you learned in (a) to the basis {qk+1 , . . . , qn } of V⊥ to compute projV⊥ b, the projection of b onto V⊥ .

(ii) Equating your answer above and the answer in (a) (v), express b as a linear combination of q1 , . . . , qn.

(iii) What are the coordinates of b in the basis q1 , . . . , qn?

(i) Find a vector u so that projection onto L is x ↦ uu⊺x.

⎛2 ⎞

(ii) Compute the projection of b = ⎜3⎟ onto L and L⊥ . Show all work. ⎝4⎠