Linear algebra questions

Math 2568 Autumn 2020

Homework 12

Problem 1 Let L : R4 → R3 be given by

L ([

x1 x2 x3 x4 ]T)

= [ (5x1 − x2 + 3x3) (6x1 + 7x2 − x4) (9x2 − x3 + 10x4)

]T • Show that L is a linear transformation.

• Find the matrix representation A of L with respect to the standard bases of R4 and R3.

• Find a basis for ker(L).

Problem 2 Let B be the matrix given by

B =

1 0 2b a c c −a b

 where a, b, c are indeterminates. Using row operations that exist for all values of a, b, or c, together with cofactor expansion, compute the determinant of B expressed as a function of a, b, and c. Use this to determine a relation between a, b, and c that provides necessary and sufficient conditions for the matrix B to be singular (your relation should be an equation involving a, b, and c).

Problem 3 Let A =

[ 2 1 −1 4

] .

a) Compute the characteristic polynomial of A.

b) For each eigenvalue of A find a basis for the corresponding eigenspace.

c) Determine if A is defective. Justify your answer.

Problem 4 Three data points are given by (−2,−7), (1, 3), (3, 5), and (7, 10).

a) Find the least-squares fit by a linear function.

b) Find the smallest degree polynomial which fits the points exactly.

Problem 5 An inner product on R2 is given on basis vectors by

< e1, e1 >= 4; < e1, e2 >=< e2, e1 >= 7; < e2, e2 >= 16

a) Find the matrix representation of the inner product.

b) If v = [3 − 2]T ,w = [4 5]T , compute < v,w >.

c) If v is as in b), find a non-zero vector u with < v,u >= 0

Problem 6 Let u1 =

 3−1 2

, u2 = 50

4

. Also let v = −32

3

. a) Compute prW (v) where W = Span{u1,u2}.

b) Compute the distance between v and W .

c) Determine the least-squares approximation of v by a vector in W .

Problem 7 Let u1 =

 −1 2 1 −2

, u2 = 

3 0 4 −3

. Let W = Span{u1,u2}. a) Use Gram-Schmit to find an orthogonal basis for W .

b) Find a matrix A with column space C(A) = W . Then use the Fundamental Subspaces Theorem to find a basis for W⊥.

c) Let v =

 2 0 −3 7

 Compute the projection prW (v) of v onto the subspace W . Then use this to compute the distance between v and W .

d) Determine the projection prW⊥(v) of v onto W ⊥.

e) Compare v to prW (v) + prW⊥(v).