Question

let n€N consider the set of nxn symmetric matrices over R with the usual addition and multiplication by a scalar. show that this set with the given operations is a vector subspace of M_nn

Answer 1

We have

$V=M_n(\R)$

is the vector space over $\R$ .

Now,

denote $A_{ij}$ is the matrix whose $(i,j)^{th}$ entry is $1$ if $i=j$

and ;if

$$i\ne j,\,\forall i,j\in \{1,\dots,n\}$$

Also denote the collection of all such matrix by $X$ ,

thus;

$W=\text{span}(X)$ is the smallest subspace

such that $I_{n\times n}\in W$ .

Define

$$M_n^s(\R):=\{A\in M_n(\R):A=A^T\}$$

Clearly, $M_n^s(\R)$ is the set of all symmetric matrix. we show that it is subspace of $M_n(\R)$ .

Let, $a,b\in \R$ and $A,B\in M_n^s(\R)$ , thus $A^T=A,B^T=B$

Now, consider the liner combination $aA+bB$ .

Note that

$$(aA+bB)^T=(bB)^T+(aA)^T\\ =(aA)^T+(bB)^T\\ =aA^T+bB^T\\ =aA+bB$$

Thus,we can see that

$$aA+bB\in M_n^s(\R)$$

Hence,it can be seen that

$M_n^s(\R)$ is subspace of $M_n(\R)$