Math 191 Wilkening
Spring 2023
Homework 4
due Sat, Feb. 18, 2:00 PM
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1. SupposeV,W andZ arefinitedimensionalvectorspacesandA:V →W andB:W →Z are linear operators.
(a) (2 points) Show that if A is surjective (onto W), then
nullity(BA) = nullity(A) + nullity(B). (b) (2 points) Show that in general (whether A is surjective or not),
nullity(BA) = nullity(A) + dim R(A) ∩ N (B).
2. (2 points) (I.5.1, page 35, Strang.) If u and v are orthogonal unit vectors, show that u + v is
orthogonal to u − v. What are the lengths of those vectors?
3. (Variant of I.5.4, page 35, Strang.) In this problem, ∥ · ∥ is the 2-norm.
(a) (2 points) Show that for x,y ∈ Rn, xTy = 1∥x+y∥2 −∥x−y∥2. This is known as the
(b) (2 points) Suppose Q is an n × n matrix with the property that for every x ∈ Rn, ∥Qx∥ = ∥x∥.
polarization identity.
Show that (Qx)T(Qy) = xTy for all x,y ∈ Rn.
4. (2 points) (I.5.5, page 35, Strang.) If Q is orthogonal, how do you know that Q is invertible and Q−1 is also orthogonal? If QT = Q−1 and QT = Q−1, show that Q Q is also an orthogonal matrix.
5. (1 points) (Variant of I.5.6, page 35, Strang.) A permutation matrix P on Rn has the same columns as the identity matrix (in some order). This means there is a permutation (bijection σ : {1,…,n} → {1,…,n}) such that
1 i = σ(j), Pij = Ii,σ(j) = 0 i ̸= σ(j).
ShowthatPTP =I,i.e.,thatP isorthogonal.
程序代写 CS代考 加QQ: 749389476
6. (2 points) (II.2.8, page 135, Strang.) What multiple of a = 1 should be subtracted from
b = 0 to make the result w2 orthogonal to a?
7. (3 points) (II.2.9, page 135, Strang.) Complete the Gram-Schmidt process in the previous problem by computing q1 = a/∥a∥ and w2 = b − (bT q1)q1 and q2 = w2/∥w2∥ and factoring into QR:
1 4 ∥a∥ ? 1 0 = q1 q2 0 ∥w2∥ .
8. (3 points) Suppose A is an m×n matrix with m ≥ n and rank n. Given b ∈ Rm, define the residual function r : Rn → Rm by
ri(x1,…,xn)=bi −aijxj, (1≤i≤m), i.e.,r(x)=b−Ax.
Compute the gradient ∇f of the objective function
ri(x1, . . . , xn)2 so f(x) = 2rT r = 2∥b − Ax∥2
f(x1, . . . , xn) = 2
and set ∇f = 0 to obtain the normal equations AT Ax = AT b for minimizing ∥b − Ax∥.
9 10 11 12
(a) (1 point) What condition on b is required for a solution of Ax = b to exist?
(b) (1 point) Find the general solution of Ax = b with b = 1 . −1
(c) (2 points) Which of the solutions x from part (b) has the smallest 2-norm?
10. SupposeAism×nwithm≥nandrankn. (a) (3 points) Show that solving
I Ar b AT 0 x = 0
yields a solution x that minimizes ∥b − Ax∥.
(b) (2 points) Derive an explicit formula for AT 0 .
1 2 3 4 9.LetA=5 6 7 8.