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 $S$ of all solutions of homogeneous equation $AX = 0$ of any $2 \times 2$ matrix $A$ and $X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2$.

b) The closed $L_2$ ball $= B(0,1) = \{ X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \in R^2 : \|X\| \leq 1 \}$. (1.5+1.5)

Define an involutory matrix. Is the matrix $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ involutory? Justify. Prove that the inverse of the transpose of a non-singular matrix $A$ is the transpose of its inverse. (0.5+1+1.5)

Define linear transformation. Let $T: R^2 \to R^2$ be defined by $T(0, 1) = (2, 1)$, $T(1, 4) = (0, -2)$. If $T$ is linear, find the formula for $T(x, y)$. Also, find the matrix represented by $T$ relative to the standard basis. (0.5+1.5+1)

What do you mean by a symmetric bilinear form on a vector space $V$ over the field $F$? Let $A$ be a symmetric matrix. Prove or disprove that the mapping $B_A(u,v) = u^T A v, \ \forall u,v \in V$ is a symmetric bilinear form on $V$. (1+2)

Let $v_1, v_2, \ldots, v_n$ be the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ of an $n \times n$ symmetric matrix $A$ respectively, then prove that

Define $L_1, L_2$ and $L_\infty$ norms on a vector $n$-space $R^n$. Let $\|.\|$ be the Euclidean norm, and $X$ & $Y$ be two vectors in $R^n$. Prove the Cauchy-Schwarz inequality $|X.Y| \leq \|X\| \|Y\|$. Verify this property for $X = (1, 3)$ and $Y = (2, 1)$. (1+4+1)

Distinguish between linear dependent and independent vectors. Prove that the linear hull of a given set of vectors $v_1, v_2, \ldots, v_n$ in a vector space $V$ is a subspace of $V$. Also, the representation of any vector in a vector space in terms of its basis vectors is unique. (1+2.5+2.5)

OR

Define a basis and dimension of a vector space. Show that the set $B = \{(1, 1, 1), (1, -1, 1), (2, 0, 3)\}$ forms a basis for $R^3$. Also, find the co-ordinates of $v = (1, 3, 2) \in R^3$ with respect to the basis $B$. (1+3+2)

Define fourier coefficient of a vector $u$ on a vector $v$. Prove that a set of non-zero orthogonal vectors is linearly independent. Also, find an orthonormal basis from the basis $\{(1, 0, 1), (1, 1, 0), (1, 1, 1)\}$ of $R^3$ using Gram Schmidt Orthogonalization Process. (1+1.5+3.5)

What are the conditions necessary for a matrix to possess an inverse? Prove that the inverse of a square matrix if it exists, is unique. If $A$, $B$, and $C$ are matrices of order $m \times n$, $n \times p$, $p \times q$ respectively, then prove that $(AB)C = A(BC)$. (1+1+4)

OR

Define 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)

Find the eigenvalues and the corresponding eigenvectors of the matrix $A = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$.

a) Diagonalize the matrix $A$. b) Find the quadratic form determined by $A$ and test its definiteness. c) Remove the cross term of the quadratic form. d) Examine the maximum and minimum value of quadratic form subject to the constraint $\|X^T X\| = 1$. (2+1.5+1+1.5)