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

The smallest subspace of $M_{n} (R)$ containing $\begin{pmatrix} 1& 0 &\cdots & 0 \\ 0 & 1 &\cdots & 0\\ \cdots & \cdots &\cdots\\ 0 & 0 & \cdots & 1\\ \end{pmatrix}$ is

$\{\lambda *$ $\begin{pmatrix} 1& 0 &\cdots & 0 \\ 0 & 1 &\cdots & 0\\ \cdots & \cdots &\cdots\\ 0 & 0 & \cdots & 1\\ \end{pmatrix}$ $| \lambda\in R\}$

$A=A^{T} ,B=B^{T} and \ c\in R$

Then (A+B)$^{T}$ $=A^{T}+B^{T}=A+B$

Again (cA)$^{T}=c *A^{T}=c A$

Thefore set of all symmetric matrices is a subspace of $M_{n}(R)$