Do 3x3 Symmetric Matrices Form a Subspace of 3x3 Matrices?

[hahatyshka]

symmetric matrices... help please!

hi can someone tell me...how to correctly use the 10 axioms..
for example: does the set of all 3x3 symmetric matrices form a subspace of a 3x3 matrices under the normal operations of matrix addition and multiplication?
I don't really get how to prove this..

 

[HallsofIvy]

hi can someone tell me...how to correctly use the 10 axioms..
for example: does the set of all 3x3 symmetric matrices form a subspace of a 3x3 matrices under the normal operations of matrix addition and multiplication?
I don't really get how to prove this..

The only things you need to prove are that the set is closed under addition and scalar multiplication. Suppose
$$A= \begin{bmatrix}a & b & c \\b & d & e \\c & e & f\end{bmatrix}$$
and
$$B= \begin{bmatrix} u & v & w \\ v & x & y \\w & y & z\end{bmatrix}$$
two symmetric matrices. Is A+ B symmetric? Is $\alpha A$ symmetric for any number $\alpha$?

 

[Legion81]

The only things you need to prove are that the set is closed under addition and scalar multiplication.

I thought a subspace had to also contain the zero vector (not empty), or am I remembering that wrong?

 

[HallsofIvy]

Well, yes. Thanks for the correction. I should have included that- but the 0 matrix is rather trivially symmetric isn't it?

My point is that you do NOT have to prove all "10 axioms". Given a subset of a vector space, things such as "u+ v= v+ u" follow from the fact that they are true of the entire vector space and so true for any subset.

To prove that "U is a subspace of vector space V" you need to prove:
1) U is closed under vector addition.
2) U is closed under scalar multiplication.
and the one I forgot:
3) U is non-empty.