Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2078 (Set Reassessment I 2078, p5-6 (Group A on p5, Group B Q6-10 across p5-6)) Nepal

Let $u = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $v = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$.

(a) Write the vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ in terms of the vectors $u$ and $v$.

(b) Show that the vectors $u$ and $v$ span $\mathbb{R}^2$.

How many kinds of subspaces of $\mathbb{R}^3$ are there? Mention them. Show that a plane through the origin is a two-dimensional subspace of $\mathbb{R}^3$.

What is the angle between the diagonal of the unit cube in the positive orthant and the vector $e_1$?

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

Let $v_1 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix}$, $v_2 = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix}$. Show that $B = \{v_1, v_2\}$ is an orthonormal basis for $\mathbb{R}^2$. Find a vector $x \in \mathbb{R}^2$ with respect to the basis $B$.

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

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

By showing that the $L_1$-norm satisfies each of the conditions in the definition of a norm prove this is a vector norm. First do this for $\mathbb{R}^2$, and then do this for $\mathbb{R}^n$.

OR

Prove that if $x \in \mathbb{R}^n$, then $\|x\|_\infty \leq \|x\|_1 \leq n\|x\|_\infty$.

Let

Show that $B$ is a basis for $V$.

OR

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

(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 $V$.

Describe the Gram-Schmidt Process to transform a basis first for $\mathbb{R}^2$ and $\mathbb{R}^3$ with illustrations and then for $\mathbb{R}^n$ to an orthonormal basis.