Can someone please guide me with this,
I want to find a basis and its dimension for the subspace of 2×2 matrices such that
$$\begin{pmatrix} 1 & 2 \\ 2 & 4 \\ \end{pmatrix} \bullet \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} =0$$
I tried this;
I know I would get the linear system
$$a+2c=0$$
$$b+2d=0$$
$$2a+4c=0$$
$$2b+4d=0$$
which equivalent leaves the two equations $a+2c=0$ and $b+2d=0$
But I don't know how to proceed from this. Would I just factor our a,b,c,d from four matrices and those would be a basis? of dim 4? Anyone can help please?
Best Answer
Notice that $a$ and $b$ determine $c$ and $d$, respectively - hence there are two free variables and the space has dimension $2$. Thus, once we find two linearly independent vectors in the space, they form a basis.
One example of that is to choose the matrix where $a = 1$ and $b = 0$, together with the matrix where $a = 0$ and $b = 1$. I'll leave it to you to check that they indeed are linearly independent.