Define augmented matrix with an example. Find the general solution of the linear system whose augmented matrix is:

\[
\begin{bmatrix}
1 & 6 & 2 & -5 \\
-1 & 0 & 3 & 1 \\
0 & -1 & -2 & 3
\end{bmatrix}
\]

​​

Let \[A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}
\]

and Define \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(x) = A x \).

Find the image under \[
T \left( u = \begin{bmatrix} 1 \\ -3 \end{bmatrix}, \, v = \begin{bmatrix} u \\ u \end{bmatrix} \right)
\]

b) Prove that a map \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T(x,y) = (x,y) \) is linear.

Define the inverse of a matrix. Find the inverse of matrix \[
A = \begin{bmatrix} 1 & 0 \\ 3 & 4 \end{bmatrix}
\] if it exists.

Verify that
\[
(AB)^{-1} = B^{-1} A^{-1}
\]
if
\[
A = \begin{bmatrix} 1 & 1 \\ 3 & 4 \end{bmatrix}, \quad
B = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}.
\]

Find LU factorization of \[ A = \begin{bmatrix} 2 & 5 \\ 6 & -7 \end{bmatrix} \]

Compute the determinant by cofactor expansions: \[
\begin{vmatrix}
6 & 0 & 0 & 5 \\
1 & 7 & 2 & -5 \\
2 & 0 & 0 & 0 \\
8 & 3 & 1 & 8
\end{vmatrix}
\]

Show that the column vectors \[
u = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad
v = \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \quad
w = \begin{bmatrix} 1 \\ 2 \\ 6 \end{bmatrix}, \quad
x = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}
\]lie in span $\{u,v,w\}$.

Let B={b1,b2}, C={c1,c2}  be bases for a vector space $V$, and suppose \[
b_1 = 6c_1 – 2c_2, \quad b_2 = 9c_1 – 4c_2
\]

(a) Find the change of coordinates matrix from $B$ to $C$.
(b) Find \([x]_C\).

\[
x = -3b_1 + 2b_2
\]

Let \[
A = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}
\] Find the eigenvalues and eigenvectors of $A$.

Determine the least squares error in the least-square solution of Ax=b, where \[
A = \begin{bmatrix} 4 & 0 \\ 0 & 2 \\ 1 & 1 \end{bmatrix}, \quad
x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad
b = \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}
\]

Prove that the binary operation ∗ defined on Z by letting
\[
m \ast n = m + n + 1
\]
is commutative and associative.

Show that \[
(\mathbb{Q}, +, \cdot)
\] forms a ring.