[SOLVED] MA502 Homework 2

1. 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.