18.06 Problem Set 3

Due Wednesday, March 13.

Problem 1. (5) Which of the following sets are bases for ℝ^{3}?

(a) {(1,2,0), (0,1,-1)}

(b) {(1,1,-1), (2,3,4), (4,1,-1), (0,1,-1)}

(c) {(3,2,2), (-1,2,1), (0,1,0)}

(d) {(1,0,0), (0,2,-1), (3,4,1), (0,1,0)}

Problem 2. (5)

(a) Let M_{2} be the vector space of 2 × 2 matrices. Determine whether or not the following set is

a basis for M_{2}: \{[ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 1 & 1 \end{matrix} ] , [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ]\}.

(b) Find a basis for the subspace of M_{2} consisting of symmetric matrices. Problem 3. (8) Find the dimensions of the following subspaces of ℝ^{4}:

(a) All vectors of the form (a, b, c, d) with d = a + b.

(b) All vectors of the form (a, b, c, d) with c = a − b and d = a + b.

(c) All vectors of the form (a, b, c, d) with a = b.

(d) All vectors of the form (a + c, a − b, b + c, −a + b). Problem 4. (6) Find a basis for each of the four subspaces associated to the matrix

A = [ \begin{matrix}0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 4 & 6 \\ 0 & 0 & 0 & 1 & 2 \end{matrix} ] .

Problem 5. (6) Suppose that {v_{1}, v_{2}, …, v_{n}} is a basis for ℝ^{n}. Show that if A is an n×n invertible matrix, then {Av_{1}, Av_{2}, …, Av_{n}} is also a basis for ℝ^{n}.