When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve the system of equations: x – 2y = 5, -x + y + 5z = 2, y + z = 0

What is the condition of a matrix to have an inverse? Find the inverse of the matrix \(A = \begin{bmatrix} 5 & 1 & 2\\ 1 & 0 & 3\\ 4 & -3 & 8 \end{bmatrix}\)  If it exists.

Find the least-square solution of Ax=b for \(A = \begin{bmatrix} 1 & -6\\ 1 & -2\\ 1 & 1\\ 1 & 7 \end{bmatrix}\) and \(b = \begin{bmatrix}-1 \\ 2\\ 1\\ 6 \end{bmatrix}\)

Let T is a linear transformation. Find the standard matrix of T such that

1. T:R² → R⁴ by T(e1) = (3, 1, 3, 1) and T(e₂) = (-5, 2, 0, 0) where e₁ = (1, 0) and e₂ = (0, 1);
2. T:R² → R⁴ rotates point as the origin through \(\frac{3\pi}{4}\) radians counter clockwise.
3. T:R² → R⁴ Is a vertical shear transformation that maps e₁ into e₁-2e₂ but leaves vector e₂ unchanged.

For what value of h will y be in span {v₁ , v₂, v₃} if \(v_1 = \begin{bmatrix}1\\ -1\\ -2\end{bmatrix}\), \(v_2 = \begin{bmatrix}5\\ -4\\ -7\end{bmatrix}\), \(v_3 = \begin{bmatrix}-3\\ 1\\ 0\end{bmatrix}\) and \(y = \begin{bmatrix}-4\\ 3\\ h\end{bmatrix}\)

Let us define a linear transformation T:R² → R² by T(x) = \(\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}\begin{bmatrix}x_1\\ x_2\end{bmatrix}\) = \(\begin{bmatrix}-x_2\\ x_1\end{bmatrix}\). Find the image under T of \(u = \begin{bmatrix}4\\ 1\end{bmatrix}\), \(v = \begin{bmatrix}2\\ 3\end{bmatrix}\) and u + v = \(\begin{bmatrix}6\\ 4\end{bmatrix}\)

Let \(A = \begin{bmatrix}2 & 5\\ -3 & 1\end{bmatrix}\) and \(B = \begin{bmatrix}4 & -5\\ 3 & k\end{bmatrix}\). Determine the value (s) of k if any will make AB = BA.

Define determinant. Compute the determinant without expanding \(\begin{bmatrix}-2 & 8 & -9\\ -1 & 7 & 0\\ 1 & -4 & 2\end{bmatrix}\)

Define null space . Find the basis for the null space of the matrix \(A = \begin{bmatrix}1 & 2 & 3\\ 2 & 3 & 4\end{bmatrix}\)

Let B = {b₁, b₂} and C = (c₁, c₂) be bases for a vector  V, and suppose b₁ = -c₁ + 4c₂ and b₂ = 5c₁ – 3c₂. Find the change of coordinate matrix for a vector space and find [x]_c for x = 5b₁ + 3b₂.

Find the eigen values of the matrix \(\begin{bmatrix}6 & 5\\ -8 & -6\end{bmatrix}\)

Find the QR factorization of the matrix \(\begin{bmatrix}2 & 1\\ 3 & -1\end{bmatrix}\)

Define binary operation. Determine whether the binary operation * is associative or commutative or both where * is defined on Q by letting \(x * y = \frac{x+y}{3}\)

Show that the ring (Z₄, +₄, .₄) is an integral domain.

Find the vector x determined by the coordinate vector \([x]_{β} = \begin{bmatrix}-4\\ 8\\ 7\end{bmatrix}\) where \( β = \left \{ \begin{bmatrix}-1\\ 2\\ 0\end{bmatrix}, \begin{bmatrix}3\\ -5\\ 2\end{bmatrix}, \begin{bmatrix}4\\ -7\\ 3\end{bmatrix} \right \}\)