Thread: basis for Snn

Another one. The M n,n is a vector space, and that Snn( the set of n x n symmetric matrices) is a subspace
Find a basis for the space of all 2 x 2 symmetric matrices.
B)Find a basis for the space of all 3 x 3 symmetric matrices
C) Determine the dimensions of the space of n x n symmetric ( Hint: this essentially a counting argument.From examining the bases in the first two parts determine what goes into the basis required.

Mnn is a space of all m x n matrices Symmetric matrix Snn are in M and is a subset of all symmetric matrices, exchanges rows and columns such as A=AT (A,B) are in Symmetric matrices such as (aij=aji or bij=bji) an example of this is A+B==aij+bij, therefore (A+B)T = (aji+bji)=(BT)+(AT) hope this helps.

a 2x2 symmetric matrix is of the form:



this suggests the basis set:



linear independence and spanning should be clear (that is: you should be able to prove these on demand).

for 3x3 symmetric matrices, prove the following is a basis set:



does this suggest what to do for the nxn case? (hint: consider the diagonal and off-diagonal entries separately).

Thank can anyone help with part C given Deveno's answer.

another hint: you should have n(n+1)/2 matrices. where did i get this number?