Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2081 (Set pages 1-2; First Assessment 2081) Nepal

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

Write the vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.

Define linearly independent vectors. Prove that an orthogonal set of nonzero vectors in a vector space is linearly independent.

Show that $u = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ -1 \end{pmatrix}$ are orthogonal in $\mathbf{R}^2$ and find corresponding orthonormal basis for $\mathbf{R}^2$.

What does "a square matrix $B$ is similar to a matrix $A$" mean? Prove that if $A$ and $B$ are similar matrices, their eigenvalues are identical.

Explain four major ways to view a matrix. Prove that if $S : \mathbf{R}^l \to \mathbf{R}^m$ and $T : \mathbf{R}^n \to \mathbf{R}^l$ are linear transformations, given by matrices $A$ and $B$, respectively, then, the composition $S \circ T : \mathbf{R}^n \to \mathbf{R}^m$ is a linear transformation and is given by $AB$.

OR

Let $S$ and $T$ be matrix transformations defined by $S(v) = Ay$ and $T(x) = Bx$, where

a) What are the domains and codomains of $S$ and $T$? Why is the composite transformation $S \circ T$ defined? What are the domain and the codomain of $S \circ T$? b) Let $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$. Determine $T(x)$. c) Find $(S \circ T)(x)$. d) Find a matrix $C$ so that $(S \circ T)(x) = Cx$. e) Show that $S \circ T$ is linear.

Define the span of a set. Prove that span $\{v_1, v_2, \ldots, v_k\} \subseteq V$ is a subspace of a vector space $V$. Also, show that if $x \in \mathbf{R}^2$, such that $x \neq 0$, then $x^{\perp}$ is a subspace of $\mathbf{R}^2$.

Let

Show that $V$ is a subspace of $\mathbf{R}^3$, and $B$ is a basis for $V$.

OR

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 $V$, derive the expression for the projection of $w$ onto $V$. b) If $\{v_1, v_2, \ldots, v_k\}$ is an orthonormal basis for $V$, derive the expression for the projection of $w$ onto $V$.

Consider the following matrix: $A = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$. a) What can we say about the action of $A$ on an arbitrary vector? b) What are examples of eigenvalues and eigenvectors of this matrix? c) What does the discussion for this example illustrate?

Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$.