Master in Data Science (SMS, TU) Mathematics for Data Science Question Paper 2079 Nepal

Show that

a. The line $x_2 = \alpha x_1$ is a subspace $\mathbb{R}^2$. b. The set of points that is the union of two lines through the origin is not a subspace.

Let

Show that $B = \{v_1, v_2\}$ is an orthonormal basis for $\mathbb{R}^2$. Find a vector $x \in \mathbb{R}^2$ with respect to the basis $B$.

Without calculation, find one eigenvalue and two linearly independent eigenvectors of

Justify your answer.

Let $Q(x) = 3x_1^2 + 9x_2^2 + 8x_1 x_2$. Find (a) the maximum value of $Q(x)$ subject to the constraint $x^T x = 1$, (b) a unit vector $u$ where this maximum is attained, and (c) the maximum of $Q(x)$ subject to the constraints $x^T x = 1$ and $x^T u = 0$.

Describe and compare the solution sets of $x_1 + 9x_2 - 4x_3 = 0$ and $x_1 + 9x_2 - 4x_3 = 2$.

Give a geometric description of span $(v)$ and span $(u, v)$. Consider the vectors $u = \begin{pmatrix}1\\2\end{pmatrix}$ and $v = \begin{pmatrix}-1\\1\end{pmatrix}$.

a. Write the vector $\begin{pmatrix}3\\2\end{pmatrix}$ in terms of the vectors $u$ and $v$. b. Show that the vectors $u$ and $v$ span $\mathbb{R}^2$.

Let $u_1 = (2\ 0\ 0)^T$, $u_2 = (0\ 1\ 1)^T$ and $u_3 = (0\ 1\ -1)^T$. Find the orthonormal set associated with the set $S = \{u_1, u_2, u_3\}$.

Prove that an orthogonal set of nonzero vectors in a vector space is linearly independent.

Consider the matrix: $A = \begin{pmatrix}3 & 0\\0 & 1\end{pmatrix}$. What can you say about the action of $A$ on an arbitrary vector? What are examples of eigenvalues/eigenvectors of this matrix? What does this discussion for this example illustrate?

OR

Let $v_1, v_2$ be the eigenvectors associated with the eigenvalues $\lambda_1, \lambda_2$ of a $2 \times 2$ symmetric matrix $A$ respectively. Prove that $A = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T$. (1)

Prove that if $B$ is a symmetric bilinear function on $\mathbb{R}^n$, then it is of the form $B = B_A(v, w) = v^T A w$, for some unique symmetric matrix $A$.

OR

Express the quadratic form $Q(x) = x_1 x_2 - x_1 x_3 + x_2 x_3$ as a sum of squares.

Find the SVD of $A = \begin{pmatrix}3 & 2 & 2\\2 & 3 & -2\end{pmatrix}$. If $A$ an invertible $n \times n$ matrix, what is the relationship between the singular values of $A$ and $A^{-1}$?