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

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

Define $L_1$, $L_2$ and $L_\infty$ norms on $\mathbf{R}^n$. Calculate $L_1$, $L_2$ and $L_\infty$ norms of the vector $x = (1, -1, 0, \ldots, 0, 2)$ on $\mathbf{R}^n$.

What is the angle between the diagonal of the unit cube in the positive orthant and the vector $e_1$ in $\mathbf{R}^3$?

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 $\mathbf{R}^2$.

Prove that eigen vectors $v_1$ and $v_2$ that correspond to distinct eigen values $\lambda_1$ and $\lambda_2$ of a $2 \times 2$ matrix are linearly independent.

Let $S$ and $T$ be matrix transformations defined by $S(y) = Ay$ 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.

Prove that if $\theta$ is the angle between two non-zero vectors $x$, $y$ in $\mathbf{R}^n$, then

Let

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

OR

Show that the following set of vectors is a basis for $\mathbf{R}^3$, and then express the standard basis vectors: $e_1$, $e_2$, $e_3$ in terms of these:

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

OR

Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 5 & -2 \\ 7 & -4 \end{pmatrix}$. State and sketch the effect of multiplying the eigen vector by the matrix $A$.

Let $A$ be a square matrix with eigen vector $u$ belonging to Eigen value $\lambda$. Prove that a) If $m$ is a natural number then $\lambda^m$ is an eigen value of the matrix $A^m$ with the same eigen vector $u$. b) If the matrix $A$ is invertible then the eigen value of the inverse matrix $A^{-1}$ is $1/\lambda = \lambda^{-1}$ with the same eigen vector $u$.