Answer
See below
Work Step by Step
We are given:
$A_1=\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
A_2=\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}\\
A_3=\begin{bmatrix}
1 & 1\\
1 & 0
\end{bmatrix}$
According to Gram-Schmidt we have:
$v_1=A_1=\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
v_2=A_2-\frac{(A_2,v_1)}{||v_1||^2}v_1\\
=\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}-\frac{(5.0.0+2.1.1+3.1.1+5.1.0)}{5.0.0+2.1.1+3.1.1+5.0.0}\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix} \\
=\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}-\frac{0+2+3+0}{0+2+3+0}\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
=A_1=\begin{bmatrix}
0 & 1\\
1 & 1
\end{bmatrix}-\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
=\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix}\\
v_3=A_3-\frac{(A_3,v_1)}{||v_1||^2}v_1-\frac{(A,v_2)}{||v_2||^2}v_2\\
=\begin{bmatrix}
1 & 1\\
1 & 0
\end{bmatrix}-\frac{5.1.0+2.1.1+3.1.1+5.0.0}{5.0.0+2.0.0+3.0.0+5.1.1}\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}-\frac{5.1.0+2.1.0+3.1.0+5.0.1}{5.0.0+2.0.0+3.0.0+5.1.1}\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix}\\
=\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix}-\frac{0+2+3+0}{0+2+3+0}\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
=\begin{bmatrix}
1 & 1\\
1 & 0
\end{bmatrix}-\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}\\
=\begin{bmatrix}
1 & 0\\
0 & 0
\end{bmatrix}$
Hence, an orthogonal basic spanned by the given factors is:
$\{\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix},\begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix},\begin{bmatrix}
1 & 0\\
0 & 0
\end{bmatrix}\}$
Determine a subspace of $M_2(R)$
Assume $\alpha, \beta, \gamma,a,b,c,d \in R$
Obtain the system:
$\alpha v_1+\beta v_2+\gamma v_3=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$
$\alpha\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}+\beta \begin{bmatrix}
0 & 0\\
0 & 1
\end{bmatrix}+\gamma \begin{bmatrix}
1 & 0\\
0 & 0
\end{bmatrix}=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}$
$\begin{bmatrix}
\gamma & \alpha\\
\alpha & \beta
\end{bmatrix}=\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\\
\rightarrow a=c$
The subspace spanned by $\{A_1,A_2,A_3\}: \{\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}: a,b,c \in R\}$
