Question

Let n$\in$N consider the set of nxn symmetric matrices over R with the usual addition and multiplication by a scalar

3.1. Show that this set with the given operations is a vector subspace of M_nn

3.2. What is the dimension of this vector subspace?

3.3. Find a basis for the vector space of 2x2 symmetric matrices.

Answer 1

1)

We depend on that $M_{n,n}$ is a vector space with respect to usual matrix operation addition and multiplication by a scalar. To prove that space $S_{n,n}$ of all symmetric matrices is a such vector space also is sufficient to prove that $S_{n,n}$ is closed for mentioned operations. Let $A,B\in S_{n,n}$ , so $\forall(i,j,1\le i,j \le n)(A_{i,j}=A_{j,i},B_{i,j}=B_{j,i})$

Let C=A+B,or $\forall(i,j,1\le i,j \le n)(C_{i,j}=A_{i,j}+B_{i,j})$

We must show that $C\in S_{n,n}$ . Let i,j be any indexes such that $1\le i,j\le n$

Then $C_{j,i}=[by\space definition ]=A_{j,i}+B_{j,i}$ But $A,B\in S_{n,n}$ , therefore

$A_{j,i}=A_{i,j},B_{j,i}=B_{i,j}=>A_{j,i}+B_{j,i}=A_{i,j}+B_{i,j}=C_{i,j}$

Thus $C\in S_{n,n}$

Further, let A be any matrix from $S_{n,n}$ and $c\in R$ any number, our task is to prove that $(cA)\in S_{n,n}$ or is symmetric. Let i,j- be any coefficients such, that $1\le i,j\le n$ , then $(cA)_{i,j}=c\cdot A_{j,i}=[A\in S_{n,n}]=c\cdot A_{i,j}=(c\cdot A)_{i,j}$ that is equivalent to $(cA)\in S_{n,n}$

2)

For linear space $S_{n,n}$ basis may be formed from elementary matrices $E^{k,m}\in S_{n,n}$ defined as $(E_{i,j}^{k,m}=1)\equiv((i=k,j=m)\lor (i=m,j=k))$

Basis in $S_{n,n}$ is $\lbrace E^{k,m}, 1\le k\le m\le n \rbrace$ with cardinalyty N=1+2+...+n=$\frac{n\cdot (n+1)}{2}$ and therefore $dim(S_{n,n})=\frac{n\cdot (n+1)}{2}$ .

if $X=(x_{i,j})_{1\le i,j\le n}$ any matrix from $S_{n,n}$ then

$X=\sum_{1\le i \le j \le n}x_{i,j}\cdot E^{i,j}$ - decomposition X by baisis.

3) If n=2 we have basis in $S_{2,2}=\lbrace \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}. \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \rbrace$