Thus far I've seen vectors and polynomials but this the first and only exercise I find that introduces matrices.
The question is as follows:
Determine whether the three matrices
$\begin{pmatrix}
1 & 1 \\
1 & 0\\ \end{pmatrix}$, $\begin{pmatrix}
-1 & 0 \\
0 & 1\\ \end{pmatrix}$, $\begin{pmatrix}
0 & 1 \\
1 & 2\\ \end{pmatrix}$span the vector space of all 2×2 symmetric matrices.
I am stuck at this stage because previously I would find the matrix of the vectors or polynomials and work on that, but this time it's 3 matrices, what are the steps that I should follow to always get it right?
Best Answer
HINT
Since we are dealing with symmetric matrices $\begin{pmatrix} a & b \\ b & c\\ \end{pmatrix}$ the dimension of the space is 3 and we can consider the equivalent vectors $(a,b,c)$. Therefore to find the dimension of the subspace spanned by the three matrices let arrange each matrix as a vector row in a 3-by-3 matrix and perform the RREF.