Reduce the system of equations into echelon form and solve:

x₁ – 2x₂ – x₃ + 3x₄ = 0

-2x₁ + 4x₂ + 5x₃ – 5x₄ = 3

3x₁ – 6x₂ – 6x₃ + 8x₄ = 2

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

The economy whose consumption matix C is

C = \(\begin{bmatrix}0.5 & 0.4 & 0.2\\ 0.2 & 0.3 & 0.1\\ 0.1 & 0.1 & 0.3\end{bmatrix}\)

and the final demand is 50 units for manufacturing, 30 units for agriculture and 20 units for service. Find the production level x that will satisfy this demand.

Find the equation y = a₀ + a₁x of the least square line that best fits the data points (0, 1), (1, 1), (1, 1), (2, 2), (3, 2).

When a linear system of equation is consistent? Find the values of h and k for which the system: 2x₁ – x₂ = h; -6x₁ + 3x₂ = k is consistent?

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

A = \(\begin{bmatrix}-2 & 8 & -1\\ 0 & 0 & 0\\ 0 & -5 & 3\end{bmatrix}\)

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 = \(\begin{bmatrix}1\\ -1\\ 3\end{bmatrix}\)

The column of I₂ = \(\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\) are (e₁) = \(\begin{bmatrix}1\\ 0\end{bmatrix}\), and e₂ = \(\begin{bmatrix}0\\ 1\end{bmatrix}\). Suppose T is a linear transformation from R² into R³ such that

T(e₁) = \(\begin{bmatrix}5\\ 1\\ -2\end{bmatrix}\) and T(e₂) = \(\begin{bmatrix}0\\ -1\\ 8\end{bmatrix}\)

find a formula for the image of an arbitrary x in R². That is, find T(x) for x in R².

Find the eigenvalues of the matrix \(\begin{pmatrix}6 & 3 & -8\\ 0 & -2 & 0\\ 1 & 0 & -3\end{pmatrix}\)

Define null space of a matrix A. If

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 null of A.

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

Evaluate the determinant of the matrix

\(\begin{bmatrix}5 & -7 & 2 & 2\\ 0 & 3 & 0 & -4\\ -5 & -8 & 0 & 3\\ 0 & 5 & 0 & -6\end{bmatrix}\)

Define unit vector. Find a unit vector v of u = (0, -2, 2, -3) in the direction of u.

Define group. Show that the set of integers is not a group with respect to subtraction operation.

Define ring. Show that set of positive integers with respect to addition and multiplication operation is not a ring.