Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2082 (Set pages 17-18; 2082 board/final exam (IOST,TU)) Nepal

What is the parallel coordinates method? Explain with an example. What is the use of this method in data science?

Let $u_1 = (1, 2, 2, -1)$, $u_2 = (1, 1, -1, 1)$, $u_3 = (-1, 1, -1, -1)$ and $B = \{u_1, u_2, u_3\}$ an orthogonal basis for $V = \text{span} \{u_1, u_2, u_3\}$. Find the projection of $w = (0, 1, 2, 3)$ onto $V$.

Let $Q(x) = 7x_1^2 + x_2^2 + 7x_3^2 - 8x_1x_2 - 4x_1x_3 - 8x_2x_3$. Find a unit vector $x$ in $\mathbf{R}^3$ at which $Q(x)$ is maximized, subject to $x^T x = 1$.

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

Describe and compare the solution sets of $x_1 + 9x_2 - 4x_3 = 0$ and $x_1 + 9x_2 - 4x_3 = 2$.

(a) Let $S : \mathbf{R}^t \to \mathbf{R}^m$ and $T : \mathbf{R}^n \to \mathbf{R}^t$ be linear transformations, given by matrices $A$ and $B$, respectively. Prove that the composition $S \circ T : \mathbf{R}^n \to \mathbf{R}^m$ is a linear transformation and is given by $AB$. (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 $\{v_1, v_2, \ldots, v_k\}$ be an orthogonal basis for $V$, then

is the projection of $w$ onto $V$. b) Moreover, if $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for $V$, then

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

OR

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) Prove that if a matrix $A$ is symmetric, then any two distinct eigenvectors corresponding to different eigenvalues are orthogonal. b) Show that the matrix $A = \begin{pmatrix} 2 & -1 \\ 1 & 4 \end{pmatrix}$ is not diagonalizable.

Find an SVD of the matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$.

OR

a) Prove that if $A$ is an $m \times n$ matrix, then all the eigenvalues of $A^T A$ are non-negative. b) Find the eigenvalues and eigenvectors of $A^T A$ where $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \\ -2 & 1 \end{pmatrix}$.

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