Finding basis and dimension of a subspace Hey can some help me with this textbook question
Let $R^{2×2}$ denote the vector space of 2×2 matrices, and let
$S =\left\{
\left[\begin{matrix}
a \space b \\ 
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$
Find (with justication) a basis for $S$ and determine the dimension of $S$.
 A: Hint: $\mathbb{R}^{2 \times 2}$ is a four-dimensional space, so the dimension of $S$ is at most $4$. In fact, it's at most $3$ since the set $S$ does not contain the matrix
$$\left[\begin{array}{c} 0 & 1 \\ -1 & 0 \end{array}\right]$$
So $S$ in fact has dimension at most $3$.

Can you find three linearly independent vectors in $S$? Note that you have three variables: $a$, $b$, and $c$; this should guide your construction.

As discussed in the comments below, three such matrices are
$$\left[ \begin{array}{c} 1 & 1 \\ 1 & 0 \end{array}\right], \left[ \begin{array}{c} 1 & 0 \\ 0 & 0 \end{array}\right], \left[ \begin{array}{c} 1 & 1 \\ 1 & 1 \end{array}\right]$$
A: Observe that
$$S = \{\left[\begin{array}{cc}a&b\\b&c\end{array}\right]: a,b,c\in\mathbb{R}\}$$
$$ = \{\left[\begin{array}{cc}a&0\\0&0\end{array}\right]+\left[\begin{array}
{cc}0&b\\b&0\end{array}\right]+\left[\begin{array}{cc}0&0\\0&c\end{array}\right]: a,b,c\in\mathbb{R}\}$$
$$ = \{a\left[\begin{array}{cc}1&0\\0&0\end{array}\right]+b\left[\begin{array}
{cc}0&1\\1&0\end{array}\right]+c\left[\begin{array}{cc}0&0\\0&1\end{array}\right]: a,b,c\in\mathbb{R}\}$$
$$ = \operatorname{span}\left(\left[\begin{array}{cc}1&0\\0&0\end{array}\right],\left[\begin{array}{cc}0&1\\1&0\end{array}\right],\left[\begin{array}{cc}0&0\\0&1\end{array}\right]\right).$$
Moreover, the list $\left(\left[\begin{array}{cc}1&0\\0&0\end{array}\right],\left[\begin{array}{cc}0&1\\1&0\end{array}\right],\left[\begin{array}{cc}0&0\\0&1\end{array}\right]\right)$ is linearly independent (why?). Hence it is a basis for $S$.
A: $S =\left\{
\left[\begin{matrix}
a \space b \\ 
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$
or [a  b  b  c]---------------(1)
Put values {[1  0  0  0 ],[0  1  0  0],[0  0  1  0],[0  0  0  1]} in (1)
we get
{[1  0  0  0 ],[0  1  1  0],[0  1  1  0],[0  0  0  1]}
$
\left[\begin{matrix}
1 \space 0 \space 0  \space 0 \\ 
0 \space1 \space 1  \space 0 \\
0 \space1 \space 1  \space 0 \\
0 \space 0 \space 0  \space 1 \\ 
\end{matrix}\right]$
I will reduce to
$
\left[\begin{matrix}
1 \space 0 \space 0  \space 0 \\ 
0 \space1 \space 1  \space 0 \\
0 \space0 \space0  \space 0 \\
0 \space 0 \space 0  \space 1 \\ 
\end{matrix}\right]$
its rank is 3,So dimension of basis of S is 3
Basis ={[1  0  0  0 ],[0  1  1  0],[0  0  0  1]} (see the reduced matrix)
Basis ={
$
\left[\begin{matrix}
1 \space 0  \\ 
0 \space0  \\
 \end{matrix}\right]$,$
\left[\begin{matrix}
0 \space 1  \\ 
1 \space0  \\
 \end{matrix}\right]$,$
\left[\begin{matrix}
0 \space 0  \\ 
0 \space1  \\
 \end{matrix}\right]$}
