Question

Linear algebra: basis

consider the subspace W={A ϵ M22(R) | [1 -1; -3 3] A=0} of M22(R). Find a basis for W.
(btw, M22 means a 2x2 matrix in this question)

How do you solve this question? What I tried doing was row reduce the matrix and got [1 -1; 0 0]. Then the solution space is [1; 1]. I think I'm doing it wrong. Can someone help me please?

Answer 1

how about
\[\left[\begin{matrix}1 & 0 \\1 & 0\end{matrix}\right]\]
and
\[\left[\begin{matrix}0 & 1 \\0 & 1\end{matrix}\right]\]

your solution found the null space, which is a single vector
\[\left(\begin{matrix}1 \\ 1\end{matrix}\right)\]

now you can form matrices like
\[\left[\begin{matrix}2 & 4 \\2 & 4\end{matrix}\right]\] as a linear combination of the 2 matrices. They are a basis for W

Answer 2

I meant to add, put your solution vector into one of the columns to form one of the matrices in the basis.

Answer 3

Oh I see. So if it asks to find a basis for M22(R) that contains the basis vectors for W that was found, do you take the basis vectors \[\left[\begin{matrix}1 & 0 \\ 1 & 0\end{matrix}\right]\] and \[\left[\begin{matrix}0 & 1 \\ 0 & 1\end{matrix}\right]\], and combine them to make a new matrix?

Answer 4

Are you asking a new question? what is a basis for all 2x2 matrices?
one answer is start with 4 zeros, and make one of them a 1. you end up with 4 2x2 matrices that can serve as a basis.
if you want to start with the 2 basis matrices in W, how about
\[\left[\begin{matrix}1 & 0\\ -1 & 0\end{matrix}\right]\]
and
\[\left[\begin{matrix}0 & 1\\ 0 & -1\end{matrix}\right]\]

Answer 5

as the other 2.

Answer 6

\[\left[\begin{matrix}1 & -1 \\ -3 & 3\end{matrix}\right]\]\[=a\left[\begin{matrix}1 & 0 \\ -1 & 0\end{matrix}\right] +b \left[\begin{matrix}0 & 1 \\ 0 & -1\end{matrix}\right]\]

then would it be something like this?

Answer 7

subspace W={A ϵ M22(R) | [1 -1; -3 3] A=0}

that means all 2x2 matrices (call them A) where
\[\left[\begin{matrix}1 & -1 \\ -3 & 3\end{matrix}\right]A=0\]

The idea is we can form all possible A's using the basis matrices you just posted.

so I would say
\[ A=a\left[\begin{matrix}1 & 0 \\ -1 & 0\end{matrix}\right] +b \left[\begin{matrix}0 & 1 \\ 0 & -1\end{matrix}\right] \]

Answer 8

Thanks so much phi :)