MA502 – Homework 2.

1. Consider T : P3 → P2 defined by differentiation, i.e., by T(p) = p

0 ∈ P2

for p ∈ P3. Find the matrix representation of T with respect to the

bases

{1 + x, 1 − x, x + x

2

, x2 + x

3} for P3 and {1, x, x2} for P2.

2. What is the dimension of S = span{ v1, v2, v3} ⊆ R

3

, where

v1 = (1, 0, 1), v2 = (1, 1, 0), and v3 = (1, −1, 2).

If the dimension is less than three, find a subset of {v1, v2, v3} that is

a basis for S and expand this basis to a basis for R

3

.

3. Consider the transformation T : R

3 → R

3 given by the orthogonal

projection onto the plane x2 = 0. (1) Find a matrix representation for

T in the coordinates induced by the canonical basis; (2) What is the

kernel of T?; (3) Find a basis for the range of T.

4. Find the matrix of transformation of coordinates (back and forth) from

the canonical basis in R

3

to the basis

B =

( 1

0

1

,

0

2

2

,

3

0

1

)

(these vectors coordinates are with respect to the canonical basis).

5. Express the linear transformation given by a clockwise rotation of π/4

in the plane spanned by e1, e2 along the e3 axis, both in terms of the

canonical basis and the basis B.

1

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