3. (a) Show that B = { 1,x+2,(x+2)
2 } and C =
{ 2,x+x2,x+1
} are bases
for P2. (b) Show that T : P2 → P2 defined by T (p(x)) = p(x+1) for each p(x) in P2 is a linear transformation. (c) Find the matrix [T ]C←B of this linear transformation T with respect to these bases B and C of P2. (d) Determine whether or not T is invertible, and if so, compute T−1.
3
4. Consider the bases B = { 1,x+2,(x+2)
2 } and C =
{ 2,x+x2,x+1
} of P2
and the polynomial q (x) = 4−3x + 2×2. (a) Find the change-of-basis matrix PC←B from the basis B to C. (b) Find the change-of-basis matrix PB←C from the basis C to B. (c) Find the coordinate vectors [q (x)]B and [q (x)]C of q (x) with respect to the bases B and C, respectively.
4
MTH 3102 Linear Algebra Final Exam May 7, 2020
Name (Last name, First name):
Exam Instructions: You have until May 8, 2020 at 10am to complete the exam. You must show your work and write in complete sentences. Partial credit may be given even for incomplete problems as long as you show your work. You are allowed to use only the book and your notes to solve the problems. You are not allowed to use any other resource.
1. Find an orthonormal basis for the column space, col(A), of the matrix A defined by
A =
1 2 1 0 −1 3 2 1 4 3 1 1
.
1
5. Let B = {v1, . . . ,vn} be an orthonormal basis for Rn. (a) Prove Parseval’s Identity, that is, for any x and y in Rn,
x ·y = (x ·v1)(y ·v1)+ · · ·+(x ·vn)(y ·vn) .
(b) What does Parseval’s Identity imply about the relationship between the dot products x ·y and [x]B · [y]B?
5
2. If b 6= 0, orthogonally diagonalize
A =
a 0 b0 a 0 b 0 a
.
2
