Finding a Basis for M2's Symmetric Matrices

[heman]

Let M2 be the vector space of 2 x 2 matrices.How to find a basis for the subspace of M2 consisting of symmetric matrices.
The problem it creates for me is that i ca guess the solution but i don't have any symstematic procedure in mind...

Pls help

 

[AKG]

For Mn, you take the n matrices that are all zeroes except have a single 1 on the diagonal, plus the n(n-1)/2 matrices that have zeroes everywhere except a 1 in the i-j position and a 1 in the j-i position, where i and j are unequal. I don't think you can get any more "systematic" than this.

 

[mathwonk]

better maybe just take your guess and try to prove it is independent and spans.

or here ios an idea: try to write down amap from some standard vector space R^t to the symmetric amtrices, in such a way that your maop is linear and an isomorphism. then it trakes a basis of the standard space to a basis of those matrices.


i.e. map say (1,0,0) to a symmetric 2by2 matrix, and (0,1,0) to another one and (0,0,1) to another one.

i.e. try mapping "upper triangular" matrices isomorphically to symmetricm ones.

 

[HallsofIvy]

Any 2 by 2 symmetric matrix must be of the form $$\begin{pmatrix}a & b \\ b & c\end{pmatrix}$$ for some numbers a, b, c.
Taking a= 1, b= c= 0 gives $$\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$$.
Taking a= 0, b= 1, c= 0 gives $$\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$$.
Taking a= b= 0, c= 1 gives $$\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$$.

Those matrices form a basis for the 3 dimensional space.

In other words, write the general matrix with constants a, b, etc. and take each succesively equal to 1, the others 0.