Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2078 (Set First Assessment 2078, p4 (Group A only fully shown; Group B Q6 begins, continues elsewhere)) Nepal

Show that

(a) The line $x_2 = ax_1$ (in usual notations, $y = ax$) is a subspace of $\mathbb{R}^2$.

(b) The line $x_2 = ax_1 + b$ (perhaps more familiar as $y = ax + b$) is not a subspace of $\mathbb{R}^2$ for $b \neq 0$.

Show that any vector in $\mathbb{R}^3$ can be expressed as a linear combination of the three unit basis vectors in $\mathbb{R}^3$. Also, show that a linear combination of the three unit basis vectors in $\mathbb{R}^3$ equals to 0 if and only if all coefficients in the linear combination are zeros.

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

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

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 $V$.

(a) By showing that the $L_\infty$-norm satisfies each of the conditions in the definition of a norm prove this is a vector norm for $\mathbb{R}^n$.

(b) Let $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$ be a vector with $x_i = i^{-1}$. Compute the 1-norm, the 2-norm, and the $\infty$-norm of $x$.

OR

Prove that if $x$ and $y$ are vectors in $\mathbb{R}^n$, then

(a) $|x \cdot y| \leq \|x\|_2 \|y\|_2$.