Consider T : P3 →P2 defined by differentiation, i.e., by T(p) = p0 ∈P2 for p ∈ P3. Find the matrix representation of T with respect to the bases
{1 + x,1 − x,x + x2,x2 + x3} for P3 and {1,x,x2} for P2.
2. What is the dimension of S = span{ v1,v2,v3}⊆R3, 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 R3.
3. Consider the transformation T : R3 → R3 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) fromthe canonical basis in R3 to the basis
(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.
