Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2081 Nepal

Show that

a) The line $x_2 = ax_1$ is a subspace $\mathbf{R}^2$. b) The line $x_2 = ax_1 + b$ is not a subspace $\mathbf{R}^2$ for $b \neq 0$.

Find the maximum value of $Q(x) = x_1^2 + x_2^2 - 10x_1x_2$ subject to the constraints $x^T x = 1$. Find a unit vector $x$ in $\mathbf{R}^2$ at which $Q(x)$ is maximized.

Find the singular values of the matrix $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \end{pmatrix}$.

Find a basis for the solution space of the equation $x_1 - y - z = 0$.

When a system of linear equations said to be consistent? Determine if the following system is consistent. Do not completely solve the system.

$x_1 + 3x_3 = 2, \quad x_2 - 3x_4 = 3, \quad -2x_2 + 3x_3 + 2x_4 = 1, \quad 3x_1 + 7x_4 = -5.$

a) Let $A$ be an $m \times n$ matrix, $B$ an $n \times p$ matrix, and $C$ a $p \times q$ matrix, so that $(AB)C$ and $A(BC)$ are defined. Prove that $(AB)C = A(BC)$. Determine the size of the matrix $A(BC)$.

b) Let $x = x_1 e_1 + x_2 e_2$, where $e_1$ and $e_2$ are unit basis vectors of $\mathbf{R}^2$. When does the equality $x = 0$ hold? What does $x = 0$ mean?

Let $V$ be a subspace of $\mathbf{R}^n$ and $w$ a vector in $\mathbf{R}^n$.

a) If $\{v_1, v_2, \ldots, v_k\}$ is an orthogonal basis for $V$, then prove that

is the projection of $w$ onto $V$.

b) Moreover, if $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for a vector space $V$, then prove that

is the projection of $w$ onto $V$.

OR

Give a geometric description of $\text{span}(v)$ and $\text{span}(u, v)$. Consider the vectors $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.

a) Write the vector $w = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u$ and $v$. b) Show that the vectors $u$ and $v$ span $\mathbf{R}^2$.

a) Let $v_1, v_2$ be the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2$ of a $2 \times 2$ symmetric matrix $A$ respectively. Prove that $A = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T$.

b) Find an orthogonal matrix $V$ which diagonalizes the matrix $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$. Also check that $V^T A V = \Lambda$, where $\Lambda$ is a diagonal matrix.

OR

a) Prove that if $A \in \mathbf{R}^{3 \times 3}$ with a quadratic form in 3 variables, then there is a symmetric matrix $B \in \mathbf{R}^{3 \times 3}$ such that

b) Classify the quadratic form: $-x_1^2 - 2x_1x_2 - x_2^2$. Then make a change of variable, $x = Py$, that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Determine $P$.

Let $A$ be an $m \times n$ matrix. Prove that

a) $A^T A$ is a square matrix. b) $A^T A$ is symmetric and so it is orthogonally diagonalizable. c) All the eigenvalues of $A^T A$ are non-negative.

What is a reduced row echelon form? Solve the following linear system by transferring the augmented matrix in reduced row echelon form.

$6x - 3y + z = 31, \quad 5x + y + 12z = 36, \quad 8x + 5y + z = 11.$