Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2080 Nepal

Are the following sets form the subspace of $R^2$? Justify.

a) The set $S = \left\{ \begin{pmatrix} x \\ y \\ z \end{pmatrix} : y + z = 2 \text{ and } x, y, z \in R \right\}$ in the vector space $R^3$.

b) The closed $L_2$ ball, $B(0,3) = \left\{ X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2 : \|X\| \le 9 \right\}$. (1.5 + 1.5)

What is rank of a matrix? Reduce the matrix $A = \begin{pmatrix} 3 & 1 & 3 & 8 \\ 2 & 2 & 6 & -1 \\ 10 & 3 & 9 & 7 \\ 8 & 4 & 11 & 17 \end{pmatrix}$ to Echelon form and hence find the rank. (1 + 2)

Let $T : R^2 \to R^2$ be defined by $T(1, 0) = (1, 3)$, $T(1, 2) = (0, -2)$. If $T$ is linear, find the formula for $T(x, y)$. What is the matrix represented by $T$ relative to the standard basis? (1.5 + 1.5)

What is meant by linear independence of a set of vectors. Let $V$ and $W$ be two vector spaces over the field $F$ and $T : V \to W$ be a linear transformation. Then prove that the kernel of $T$ is a subspace of $V$ and the image of $T$ is a subspace of $W$. (1 + 1 + 1)

What is quadratic form? Let $S$ be a $3 \times 3$ square matrix with a quadratic form in 3 variables. Then there exists a $3 \times 3$ symmetric matrix $T$ such that $X^T S X = X^T T X, \forall X \in R^3$. (1 + 2)

What is $L_\infty$ norm on a vector $n$-space $R^n$? Write any two properties. Let $\|.\|$ be the Euclidean norm, $X$ and $Y$ be two vectors in $R^n$. State and prove the Triangle Inequality and Parallelogram Law. Verify Cauchy-Schwarz inequality for $X = (1, 3)$ and $Y = (2, 1)$. (1 + 2 + 2 + 1)

Define Kernel and Image of a linear transformation $T : V \to W$. If $v_1, v_2, \ldots, v_n$ are linearly independent vectors in $V$ and $\operatorname{Ker} T = \{0_V\}$. Is the set $\{ T(v_1), T(v_2), \ldots, T(v_n) \}$ forms linearly independent vectors in $W$? Justify. Also, show that the set $\{(0, 1), (1, 1)\}$ of vectors span $R^2$. (1 + 2.5 + 2.5)

OR

What is quadratic form? Let $A$ be a $3 \times 3$ square matrix with a quadratic form in 3 variables. Then there exists a $3 \times 3$ symmetric matrix $B$ such that $X^T A X = X^T B X, \forall X \in R^3$. Further, express the quadratic form $x_1^2 + x_1 x_2 - 4 x_3 x_1 + 2 x_2 x_3 - 4 x_3^2$ as the difference of squares. (1 + 2.5 + 2.5)

Let $A = \begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix}$ be a symmetric matrix. Find the orthogonal matrix $D$ such that $D^{-1} A D$ is a diagonal matrix. Let $u_1, u_2, \ldots, u_n$ be the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ of a $n \times n$ symmetric matrix $B$ respectively, then prove that $B = \lambda_1 u_1 u_1^T + \lambda_2 u_2 u_2^T + \ldots + \lambda_n u_n u_n^T$. (3 + 3)

Explain the role of basic linear algebra techniques that are useful in the study of data science. Describe how the various concepts of Vector Spaces are applied in Machine Learning. (2 + 4)

OR

Define four fundamental subspaces of a matrix A. Determine the values of the constants $a$ and $b$ for which the system $3x - 2y + z = b$, $5x - 8y + 9z = 3$, $2x + y + az = -2$ has a unique solution, no solution and infinitely many solutions. (2 + 4)

Discuss in brief about singular value decomposition? Find the singular value decomposition of the matrix $\begin{pmatrix} 3 & 1 & 1 \\ -1 & 3 & 1 \end{pmatrix}$. (1 + 5)