18.06 Problem Set #4

Due Wednesday, Mar 20

1 Without computing A, find a basis (expain your answers) for each of the four subspaces associated with

A = [ \begin{matrix}1 & 0 & 0 \\ −4 & 1 & 0 \\ 8 & 3 & 1 \end{matrix} ] [ \begin{matrix}0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 1 & −1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ] .

2 a) Suppose the columns of A are unit vectors, all mutually perpendicular. What is A^{T} A?

b) Square matrices that satisfy the conditions in a) are called orthogonal

matrices. Prove that [ \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ] is a 2x2 orthogonal matrix

for any real number θ.

c) Prove the set of 2x2 orthogonal matrices is not a vector space by giving counter examples.

d) Show that an n × n matrix A is orthogonal if and only if A^{T} = A^{−1}.

3 Let S be the vector space spanned by vectors (1, 0, −2, 0), (0, 2, 4, −1) and (2, 2, 0, −1). Find S^{⊥}, (S^{⊥})^{⊥}. What is the relationship between (S^{⊥})^{⊥} and S?

4 Find a basis for the vector subspace S, which is the intersection of U and V where U is the span of {(1, −2, 0, 3), (0, 1, 0, −1)}, and V is the span of {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)}.

5 Find the projection of b onto the column space of A by solving A^{T} Ax̂ = A^{T} b and p = Ax̂.

a) A = [ \begin{matrix}1 & 0 \\ −2 & 1 \\ 1 & 3 \end{matrix} ] , b = [ \begin{matrix}2 \\ 3 \\ 0 \end{matrix} ]. b) A = [ \begin{matrix}0 & 2 \\ −1 & 3 \\ 2 & −1 \\ 0 & 1 \end{matrix} ] , b = [ \begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix} ]

6 Let A be an m × n matrix of rank n. Suppose that P is the projection matrix that projects onto the column space of A. What is the size of the matrix P? What is its rank? Explain.