Description
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.
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