Question 1

- **Note:** A vector $\begin{bmatrix} \bullet \\ \bullet \end{bmatrix}$ could be represented as $(\bullet, \bullet)$ or $[\bullet, \bullet]$. - $u \cdot v$ is the standard inner product of the two vectors $u$ and $v$. - $\overline{u}$ is the complex conjugate of the vector $u$. - $\|u\| = \sqrt{u \cdot u}$ - $S^{\perp}$ is the dual of the subspace $S$. - $T^*$ is the adjoint of the operator $T$. - $A^H$ is the Hermitian of the matrix $A$. **1 Objective Questions ( 20 marks)** 1. Let $A$ be a circulant matrix. Which of the following options is/are true? [4 marks] (a) If $v_1$ and $v_2$ are the eigenvectors of $A$ with different eigenvalues, then they are orthogonal. (b) If $\|Av\| = 5$ for some $v$, then $\|A^Hv\| = 5$. (c) $A$ is not a normal matrix. (d) $A^H = A$ 2. The system of differential equations $$v_1''(t) = 2\omega_0^2 v_1(t) + 3\omega_0^2 v_2(t)$$ $$v_2''(t) = -3\omega_0^2 v_1(t) - 4\omega_0^2 v_2(t)$$ can be represented as $$\begin{bmatrix} v_1''(t) \\ v_2''(t) \end{bmatrix} = \omega_0^2 A \begin{bmatrix} v_1(t) \\ v_2(t) \end{bmatrix}.$$ Which of the following options is/are true? [4 marks] (a) $A$ is negative semi-definite. (b) Rank of $A$ is 2. (c) $A$ is positive definite. (d) $A$ is negative definite. Table of 2-dimensional data (x, y) within questions 3 to 5 on eigenvalues and covariance **2 Subjective Questions ( 30 marks)** 1. Consider the vectors $u = (1,-2,1)$, $v = (3,1,-3)$, and $w = (0,1,1)$. (a) Find the projection of $u$ onto $v$. [4 marks] (b) Perform the Gram-Schmidt orthonormalization process on $u$, $v$, and $w$. [6 marks] 2. Consider a vector $v = \begin{bmatrix} 2 \\ 1 \\ -2 \\ 4 \\ -3 \end{bmatrix}$. (a) Find a circulant matrix $A$, where the first column is the vector $v$. [Marks: 2] (b) Find the length-5 DFT of the vector $v$. [Marks: 6] 3. Consider two signals $x(t) = 1 - \frac{3}{2}t$ and $y(t) = \begin{cases} 1, & 0 \leq t \leq 0.5, \\ 2, & \text{otherwise} \end{cases}$ in $L^2([0,1])$. (a) Find $||x(t)||$. [3 marks] (b) Find the inner product $\langle x(t), y(t) \rangle$. [3 marks] (c) Find the Fourier series for the signal $y(t)$. [6 marks]