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

Prove that any two eigenvectors of a symmetric matrix corresponding to different eigenvalues are orthogonal.

Define the rank of a matrix. Reduce the matrix $A = \begin{pmatrix} 6 & 1 & 3 & 8 \\ 4 & 2 & 6 & -1 \\ 10 & 3 & 9 & 7 \\ 16 & 4 & 12 & 15 \end{pmatrix}$ to Echelon form and hence find the rank. [1+2]

Under what circumstance a system of homogeneous linear equations has a non-zero solutions? Find the values of $k$ such that the system of homogeneous equations

has non-zero solutions. [1+2]

State the conditions for the various types of definiteness of a quadratic form? For what value of $\alpha$ a quadratic form $Q(x) = \alpha x_1^2 - 6x_1 x_2 + \alpha x_2^2$ is positive semi definite? [1+2]

Find the inverse of the matrix $A = \begin{bmatrix} 2 & 4 & 3 \\ 0 & 1 & 1 \\ 2 & 2 & -1 \end{bmatrix}$ by elementary row transformation.

Define a bilinear form. Give an example. Given a symmetric bilinear map $B: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ defined by $B(u, v) = B_A(u, v) = u^T A v$ for all $u, v \in \mathbb{R}^n$ for some symmetric matrix $A$, then $A$ is unique. [1+1+4]

Under what circumstance a system of homogeneous linear equations has a non-zero solutions? Determine the values of $\alpha$ and $\beta$ for which the system

has a unique solution, no solution and infinitely many solutions. [1+5]

OR

Under what circumstance a system of linear equations has a unique solution, no solution and infinitely many solutions? Find the values of $k$ such that the system of homogeneous equations $x + ky + 3z = 0, \quad 4x + 3y + kz = 0, \quad 2x + y + 2z = 0$ has non-trivial solutions. [2+4]

Explain in brief the purpose of the inclusion of Linear Algebra in the Data Science curriculum? Describe various techniques Spectral theories are applied in Data Science and their works. [2+4]

OR

Describe the role of basic linear algebra techniques which are useful in the study of data science. Describe how the various concepts of System of Linear Equations are applied in Machine Learning. [2+4]

What is singular value decomposition? Find the singular value decomposition of the matrix $\begin{pmatrix} 0 & 2 \\ 1 & -2 \\ 1 & 1 \end{pmatrix}$ [1+5]

Define four fundamental subspaces of a matrix A. Find the null space of the matrix $A = \begin{pmatrix} 2 & 3 \\ 8 & 12 \end{pmatrix}$. Find the quadratic form determined by $A$ and remove the cross term of the Quadratic form. [1.5+1.5+1.5+1.5]