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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.
(c) Compute the derivatives of the elements in the monomial basis of R[x]≤d.
(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.
(c) Based on what you just did, how would you convert a diagonalization 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.
(c) Find two rank one matrices whose sum is still rank 1 and two rank one matrices whose sum has rank 2.
(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?
(c) (4.1, #17) Let L be the line spanned by (1, 1, 1)⊺ .
(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⎠