Mathematics II TU Board 2079 question paper

Reduce the system of equations into echelon form and solve:

x_1 – 2x_2 – x_3 + 3x_4 = 0

-2x_1 + 4x_2 + 5x_3 – 5x_4 = 3

3x_1 – 6x_2 – 6x_3 + 8x_4 = 2

Define linear transformation with an example.

Let A = $\left[\begin{array}{cc}1& -3\\ 3& 5\\ -1& 7\end{array}\right]$, v = $\left[\begin{array}{c}2\\ -1\end{array}\right]$, b = $\left[\begin{array}{c}3\\ 2\\ 4\end{array}\right]$, x = $\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

and define a transformation T:R^2 → R^2by T(x) = Ax then

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

The economy whose consumption matix C is

C = $\left[\begin{array}{ccc}0.5& 0.4& 0.2\\ 0.2& 0.3& 0.1\\ 0.1& 0.1& 0.3\end{array}\right]$

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 = a_0 + a_1x 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: 2x_1 – x_2 = h; -6x_1 + 3x_2 = k is consistent?

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

A = $\left[\begin{array}{ccc}-2& 8& -1\\ 0& 0& 0\\ 0& -5& 3\end{array}\right]$

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

u = $\left[\begin{array}{c}1\\ -3\\ 2\end{array}\right]$, v = $\left[\begin{array}{c}1\\ -1\\ 3\end{array}\right]$

The column of I_2 = $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ are (e_1) = $\left[\begin{array}{c}1\\ 0\end{array}\right]$, and e_2 = $\left[\begin{array}{c}0\\ 1\end{array}\right]$. Suppose T is a linear transformation from R^2 into R^3 such that

T(e_1) = $\left[\begin{array}{c}5\\ 1\\ -2\end{array}\right]$ and T(e_2) = $\left[\begin{array}{c}0\\ -1\\ 8\end{array}\right]$

find a formula for the image of an arbitrary x in R^2. That is, find T(x) for x in R^2.

Find the eigenvalues of the matrix $\left(\begin{array}{ccc}6& 3& -8\\ 0& -2& 0\\ 1& 0& -3\end{array}\right)$

Define null space of a matrix A. If

A = $\left[\begin{array}{ccc}-1& -3& 2\\ -5& -9& 1\end{array}\right]$, and v = $\left[\begin{array}{c}5\\ -3\\ -2\end{array}\right]$

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

$\left[\begin{array}{cccc}5& -7& 2& 2\\ 0& 3& 0& -4\\ -5& -8& 0& 3\\ 0& 5& 0& -6\end{array}\right]$

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.