Define system of linear equations. When a system of equation is consistent? Determine if the system -2x₁ – 3x₂ + 4x₃ = 5, x₂ – 2x₃ = 4, x₁ + 3x₂ – x₃ = 2 is consistent.

Define linear transformation with an example.

Let A = \(\begin{bmatrix}1 & -3\\ 3 & 5\\ -1 & 7\end{bmatrix}\), v = \(\begin{bmatrix}2\\ -1\end{bmatrix}\), b = \(\begin{bmatrix}3\\ 2\\ 4\end{bmatrix}\), x = \(\begin{bmatrix}x_1\\ x_2\end{bmatrix}\)

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

1. find T(v)
2. find x ∈ R² whose image under T is b

Find the LU factorization of \(\begin{bmatrix}2 & 4 & -1 & 5 & -2\\ -4 & -5 & 3 & -8 & 1\\ 2 & -5 & -4 & 1 & 8\\ -6 & 0 & 7 & -3 & 1\end{bmatrix}\)

Find a least square solution of the inconsistent system Ax =  b for

A = \(\begin{bmatrix}-1 & 2\\ 2 & -3\\ -1 & 3\end{bmatrix}\), b = \(\begin{bmatrix}4\\ 2\\ 1\end{bmatrix}\)

Determine the column of the matrix A are linearly independent, where

\(A = \begin{bmatrix}0 & 1 & 4\\ 1 & 2 & -1\\ 5 & 8 & 0\end{bmatrix}\)

When two column vector in R² are equal? Give an example. Computer u + 3v, u – 2v, where

u = \(\begin{bmatrix}1\\ -3\\ 2\end{bmatrix}\), v = \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

Let A = \(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}\) and define T:R² → R² by T(x) = Ax, find the image under T of

\(u = \begin{bmatrix}1\\ -3\end{bmatrix}\) and \(v = \begin{bmatrix}1\\ 5\end{bmatrix}\)

Find the eigen value of \(\begin{bmatrix}3 & 6 & -8\\ 0 & 0 & 6\\ 0 & 0 & 2\end{bmatrix}\)

Define null space of a matrix A. Let

A = \(\begin{bmatrix}-1 & -3 & 2\\ -5 & -9 & 1\end{bmatrix}\), and v = \(\begin{bmatrix}5\\ -3\\ -2\end{bmatrix}\)

Then show that v is in the null A

Verify that 1^k, (-2)^k, 3^k are linearly independent signals.

If A = \(\begin{bmatrix}7 & 2\\ -4 & 1\end{bmatrix}\). find a formula for Aⁿ, where A = PDP^-1

P = \(\begin{bmatrix}1 & 1\\ -1 & -2\end{bmatrix}\) and D = \(\begin{bmatrix}5 & 0\\ 0 & 3\end{bmatrix}\)

Find a unit vector v of u = (1, -2, 2, 3) in the direction of u.

Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v

Let an operation * be defined on Q⁺ by \(a * b = \frac{ab}{2}\). Then show that Q⁺ forms a group.

Define ring and show that set of real numbers with respect to addition and multiplication operation is a ring.