Mathematics II TU Board 2078 question paper

Define system of linear equations. When a system of equation is consistent? Determine if the system -2x_1 – 3x_2 + 4x_3 = 5, x_2 – 2x_3 = 4, x_1 + 3x_2 – x_3 = 2 is consistent.

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

Find the LU factorization of $\left[\begin{array}{ccccc}2& 4& -1& 5& -2\\ -4& -5& 3& -8& 1\\ 2& -5& -4& 1& 8\\ -6& 0& 7& -3& 1\end{array}\right]$

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

A = $\left[\begin{array}{cc}-1& 2\\ 2& -3\\ -1& 3\end{array}\right]$, b = $\left[\begin{array}{c}4\\ 2\\ 1\end{array}\right]$

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

$A=\left[\begin{array}{ccc}0& 1& 4\\ 1& 2& -1\\ 5& 8& 0\end{array}\right]$

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

Let A = $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ and define T:R^2 → R^2 by T(x) = Ax, find the image under T of

$u=\left[\begin{array}{c}1\\ -3\end{array}\right]$ and $v=\left[\begin{array}{c}1\\ 5\end{array}\right]$

Find the eigen value of $\left[\begin{array}{ccc}3& 6& -8\\ 0& 0& 6\\ 0& 0& 2\end{array}\right]$

Define null space of a matrix A. Let

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 in the null A

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

If A = $\left[\begin{array}{cc}7& 2\\ -4& 1\end{array}\right]$. find a formula for A^n, where A = PDP^-1

P = $\left[\begin{array}{cc}1& 1\\ -1& -2\end{array}\right]$ and D = $\left[\begin{array}{cc}5& 0\\ 0& 3\end{array}\right]$

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 $a\ast b=\frac{ab}{2}$. 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.