Question

Consider the real vector space Mn(R), of all
n x n matrices with entries from the set of
real numbers with respect to the usual
addition and scalar multiplication of
matrices. Find the smallest subspace of
Mn(R) which contains the identity matrix.
Also show that the set of all symmetric
matrices is a subspace of Mn(R).

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 linear combination $aA+bB$ .

Note that

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

Thus,

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

Hence, $M_n^s(\R)$ is subspace of $M_n(\R)$ .