Question 1

**1 Objective Questions** 1. Consider the linear operator $T(x,y,z) = (x + 2y + 2z, 2x + y + 2z, 2x + 2y + z)$. Which of the following options is (are) true? [4 Marks] (a) $\text{Null}(T) = 0$ (b) $T^2 - 4T - 7 = 0$ (c) $T^{-1}(x,y,z) = 0.2(-3x + 2y + 2z, 2x - 3y + 2z, 2x + 2y - 3z)$ (d) Inverse of $T$ does not exist. 2. Perform Gramm-Schmidt orthonormalization on the vectors $u_1 = (1,1,0)$, $u_2 = (1,0,1)$, and $u_3 = (0,1,1)$. Which of the following options correctly represents the orthonormal basis obtained? [4 marks] (a) $\left\{\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{6}}(1,-1,2), \frac{1}{\sqrt{3}}(-1,1,-1)\right\}$ (b) $\left\{\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{3}}(1,-1,1), \frac{1}{\sqrt{6}}(-1,1,1)\right\}$ (c) $\left\{\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{6}}(1,-1,2), \frac{1}{\sqrt{3}}(-1,1,1)\right\}$ (d) $\left\{\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{3}}(1,-1,1), \frac{1}{\sqrt{6}}(1,1,-2)\right\}$ 3. Consider a graph $G$ with 4 vertices $\{V_1, V_2, V_3, V_4\}$ and edges $(V_1, V_2), (V_2, V_3), (V_3, V_4)$ and $(V_4, V_1)$. Let $A$ be the adjacency matrix of the graph $G$. Which of the following options is (are) true? [4 marks] (a) $A = \begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{bmatrix}$ (b) $2$ and $-1$ are eigenvalues of $A$. (c) $G$ is a bipartite graph. (d) $A$ is not diagonalizable. 4. Consider a quadratic form $Q(x,y) = v^TAv = 2x^2 + 2xy + 2y^2$, where $A$ is a $2 \times 2$ square symmetric matrix of order 2 and $v = \begin{bmatrix} x \\ y \end{bmatrix}$. Let $T$ be the linear operator represented by $A$. Which of the following options is (are) true? [4 marks] (a) $Q(x,y) = 1$ represents an hyperbola. (b) $T \neq T^*$ (c) The eigenvalues of $T$ are 1 and 3. (d) $\text{Rank}(A) = 2$ 5. Consider the vector $v = \begin{bmatrix} 1 \\ 3 \\ -2 \\ 4 \end{bmatrix}$ in the time domain. Which of the following vectors is the DFT of the vector $v$? [4 marks] (a) $\begin{bmatrix} 6 \\ 3+i \\ -8 \\ 3-i \end{bmatrix}$ (b) $\begin{bmatrix} 6 \\ 3-i \\ -8 \\ 3+i \end{bmatrix}$ (c) $\begin{bmatrix} 6 \\ 3+i \\ 8 \\ 3-i \end{bmatrix}$ (d) $\begin{bmatrix} 6 \\ 3-i \\ 8 \\ 3+i \end{bmatrix}$ **2 Subjective Questions** 1. Determine whether the given equations describe an ellipse. If they do, find the centre, the major and minor axes, and the major and minor radii. (i) $x^2 + 4xy + 2y^2 = 6$ [5 marks] (ii) $3x^2 - 2xy + 3y^2 = 4x + 4y$ [5 marks] 2. Consider a circulant matrix $A$ such that the first row of $A$ is $\begin{bmatrix} 0 & 6 & 7 & 8 \end{bmatrix}$. (a) Find the matrix $A$. [2 Marks] (b) Find the eigenvalues of $A$. [4 marks] (c) Find a diagonal matrix $D$ and an Unitary matrix $U$ such that $A = UDU^T$. [4 marks] 3. Consider two signals $x(t) = t$ and $y(t) = 1 - 2t^2$ in the interval $[0, 1]$. (i) Find the value of $45\|x\|^2\|y\|^2$. [4 marks] (ii) Find the inner product $\langle x, y \rangle$. [2 marks] (iii) Let $h(t) = t^2$ be a signal in the interval $[0, 1]$. Find a signal in the subspace $\text{Span}(x(t), (y(t))$ that best approximates the signal $h(t)$. [4 marks] **Hint:** Find the projection of $h(t)$ on to the subspace $\text{Span}(x(t), y(t))$.