TRIBHUVAN UNIVERSITY
INSTITUTE OF SCIENCE AND TECHNOLOGY
2080 B.S.
Candidates are required to give their answers in their own words as far as practicable.
Group A
Long Answer QuestionsAttempt any Two questions.
- 1What is a system of linear equations? When the system is consistent? Find a condition on g, h, k that makes the system consistent $x_{1}-4x_{2}+7x_{3}=g$ $3x_{2}-5x_{3}=h$ $-2x_{1}+5x_{2}-9x_{3}=k$.
- 2Let $A = \begin{bmatrix} 1 & -5 & -7 \\ -3 & 7 & 5 \end{bmatrix}$, $u = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, $b = \begin{bmatrix} -2 \\ -2 \end{bmatrix}$. Define $T : \mathbb{R}^{3} \to \mathbb{R}^{2}$ 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?
- 3Find 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.
Group B
Short Answer QuestionsAttempt any Eight questions.
- 4Are 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.
- 5Find LU factorization of $\begin{bmatrix} 2 & 3 & 4 \\ 4 & 5 & 10 \\ 4 & 8 & 2 \end{bmatrix}$
- 6Compute Det A where $A = \begin{bmatrix} 2 & -8 & 6 & 8 \\ 3 & -9 & 5 & 10 \\ -3 & 0 & 1 & -2 \\ 1 & -4 & 0 & 6 \end{bmatrix}$
- 7Show that $H = \{(a-3b, b-a, a, b) : a, b \in \mathbb{R}\}$ is a subspace of $\mathbb{R}^{4}$.
- 8Is $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ an eigenvector of $\begin{bmatrix} 5 & -3 \\ -4 & 9 \end{bmatrix}$? If so, find the eigenvalue.
- 9Let $u = (1, -2, 2, 0)$. Find a unit vector v in the same direction as u.
- 10Find the basis and dimension of Nul A where $A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 7 & 8 \end{bmatrix}$
- 11Define group. Show that $(\mathbb{Z}, .)$ doesn't form a group.
- 12Show that every field is an integral domain.