What is a system of linear equations ? When the system is consistent ? Find the condition on g, h, k that makes the system consistent.

x₁ – 4x₂ + 7x₃ = g

3x₂ – 5x₃ = h

-2x₁ + 5x₂ – 9x₃ = k

$$ Let A =\begin{bmatrix}
1 & -5 & -7 \\
-3 & 7 & 5 \\
\end{bmatrix} , u = \begin{bmatrix}
1 \\
2\\
3\\
\end{bmatrix}, b= \begin{bmatrix}
-2\\
-2\\
\end{bmatrix},$$

and define a transformation T : R³ → R² by T(x) = Ax then

a. find T(u).

b. Find x ∈ R³ whose image under T is b.

c. Is x unique ?

Find the least square solution of Ax=b where

$$ A=\begin{bmatrix}
1 & -3 & -3\\
1 & 5 & 1\\
1 & 7 & 2\\
\end{bmatrix}, b= \begin{bmatrix}
5\\
-3\\
-5\\
\end{bmatrix}$$

and compute the associated least square error.

Are vectors $$v_1=\begin{bmatrix}1\\4\\0\\ \end{bmatrix}, v_2=\begin{bmatrix}10\\2\\1\\ \end{bmatrix} and v_3=\begin{bmatrix}-5\\0\\6\\ \end{bmatrix}$$ linearly independent? Justify.

Find LU Factorization. Given the matrix:
$$
\begin{bmatrix}
2 & 3 & 4 \\
4 & 5 & 10 \\
4 & 8 & 2
\end{bmatrix}
$$

Compute Det of A where
$$
A = \begin{bmatrix}
2 & -8 & 6 & 8 \\
3 & -9 & 5 & 10 \\
-3 & 0 & 1 & -2 \\
1 & -4 & 0 & 6
\end{bmatrix}
$$

Show that H = {(a−3b, b−a, a, b) : a, b ∈ R} is a subspace of  R⁴.

Is $$\begin{bmatrix}3\\2\\ \end{bmatrix}$$ an eigen vector of  $$\begin{bmatrix}5&-3\\-4&9\\ \end{bmatrix}$$ ? If so, find eigenvalue.

Let u = (1, -2, 2, 0). Find a unit vector of  v  in the same direction of u.

Find the basis and dimension of Nul A where A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 \end{bmatrix}.

Define group. Show that (Ζ , .) doesn’t form a group.

Show that every field is an integral domain.