CSIT 2nd Semester
Mathematics II Board Question Paper 2080

What is a system of linear equations? When the system is consistent? Find a condition on \( g \), \( h \), \( k \) that makes the system consistent
\[ x_1 - 4x_2 + 7x_3 = g\]
\[ 3x_2 - 5x_3 = h\]
\[ -2x_1 + 5x_2 - 9x_3 = k.\]

Let \( A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix}, u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, b = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \). Define \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) by \( T(x) = Ax \)
(a). Compute \( T(u) \)
(b). Find a vector \( x \in \mathbb{R}^3 \) whose image under \( T \) is \( b \).
(c) Is \( x \) unique?

Find a least square solution of \( Ax = b \) where \( A = \begin{bmatrix} 1 & -3 & -3 \\ 1 & 5 & 1 \\ 1 & 7 & 2 \end{bmatrix}, b = \begin{bmatrix} 5 \\ -3 \\ -5 \end{bmatrix} \). Also, compute the associated least square error.

Are the vectors \( v_1 = \begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix}, v_2 = \begin{bmatrix} 10 \\ 2 \\ 1 \end{bmatrix}, v_3 = \begin{bmatrix} -5 \\ 0 \\ 6 \end{bmatrix} \) linearly independent? Justify.

Find LU factorization of \(\begin{bmatrix} 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 \end{bmatrix}\)

Compute \(\text{Det } A\) where \(A = \begin{bmatrix} 2 & -8 & 6 & 8 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{bmatrix}\).

Is \(\begin{bmatrix} 3 \\ 2 \end{bmatrix}\) an eigenvector of \(\begin{bmatrix} 5 & -3 \\ -4 & 9 \end{bmatrix}\)? If so, find the eigenvalue.

Let \(u = (1, -2, 2, 0)\). Find a unit vector \(v\) in the same direction as \(u\).

Find the basis and dimension of \(\text{Nul } A\) where \(A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 \end{bmatrix}\).

Define group. Show that \((\mathbb{Z},.)\) doesn't form a group.

Show that every field is an integral domain.