CSIT 2nd Semester
Mathematics II Board Question Paper 2079

Define system of linear equations. When a system of equations in consistent? Make echelon form to solve:
\[-2x_1 - 3x_2 + 4x_3 = 5 \]
\[x_2 - 2x_3 = 4 \]
\[x_1 + 3x_2 - x_3 = 2 \]
is consistent.

Define linear transformation with an example.
\[\text{Let } A = \begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}, \quad v = \begin{bmatrix} -2 \\ 1 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \]
and define a transformation \( T : \mathbb{R}^2 \to \mathbb{R}^3 \) by \( T(x) = Ax \) then
(a) find \( T(v) \)
(b) find \( x \in \mathbb{R}^2 \) whose image under \( T \) is \( b \).

Find \( AB \) by block multiplication of the matrices
\[A = \begin{bmatrix} 2 & -3 & 1 & 0 & -4 \\ 1 & -5 & -2 & 3 & -1 \\ 0 & -4 & -2 & 7 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 6 & 4 \\ 2 & -1 \\ -3 & 7 \\ 1 & 3 \\ 5 & -3 \end{bmatrix} \]

Find the least square solution of \( Ax = c \), where
\[A = \begin{bmatrix} 1 & -3 & -3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix}, \quad c = \begin{bmatrix} 5 \\ -3 \\ 5 \end{bmatrix} \]
and compute the associated least square error.

Determine the column of the matrix \( A \) are linearly independent, where
\[A = \begin{bmatrix} 3 & -3 & 6 \\ 0 & 2 & 4 \\ 0 & 3 & 0 \end{bmatrix} \]

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

When two column vectors in \( \mathbb{R}^2 \) are equal? Given an example. Compute \( u + 3v, -u - 2v \), where
\[u = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, v = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}.\]

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 \( v \).

Find the eigenvalue of \( A = \begin{bmatrix} 7 & 3 \\ 3 & -1 \end{bmatrix} \).

Define null space of a matrix \( A \). Let
\[A = \begin{bmatrix} 1 & -3 & 2 \\ -5 & 9 & -1 \end{bmatrix}, \text{and } v = \begin{bmatrix} 5 \\ 3 \\ 2 \end{bmatrix},\]
then show that \( v \) belongs to the null space matrix \( A \).

Find the equation \( y = a_0 + a_1 x \) of the least squares line that best fits the data points (2, 1), (5, 2), (7, 3), (8, 3)