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 (16 marks)

1. Consider a $5 \times 7$ matrix $A$ and a $7 \times 5$ matrix $B$. Which of the following option(s) is(are) true? [4 Marks]

(a) $\text{Rank}(AB) = \text{Rank}(A) + \text{Rank}(B)$

(b) $\text{Rank}(5A) = \text{Rank}(A)$.

(c) If the nullity of the matrix $AB$ is 2, then the rank of the matrix $AB$ is 3.

(d) $\text{Rank}(5A) \leq 5$.

1. Let $V$ be the subspace of $\mathbb{R}^3$ defined as follows:

$V = \{(x,y,z) \mid x = y - z, \text{ and } x,y,z \in \mathbb{R}\}.$

Which of the following option(s) is(are) true? [4 marks]

(a) The vectors $(1,1,0), (1,0,-1)$ in $V$ are linearly dependent.

(b) The vectors $(1,1,0), (1,0,-1), (0,1,1)$ in $V$ are linearly independent.

(c) $\text{Span}((0,1,1), (1,0,-1)) = V$.

(d) $\text{Span}((1,1,0))$ is a subspace of $V$.

1. Consider the vectors $a = (1,2)$ and $b = (2,2)$. Find the value of $||a + b|| - ||a - b||$. [4 marks]

2. Consider a linear operator $T(x,y,z) = (2x + iy, y - 5iz, x + (1-i)y + 3z)$. Let $A$ be the matrix representation of $T$. Which of the following option(s) is(are) correct? [4 marks]

(a) $T^* = T$

(b) $A^H = \begin{bmatrix} 2 & 0 & 1 \\ -i & 1 & 1+i \\ 0 & 5i & 3 \end{bmatrix}$

(c) $T \circ T^*$ is a self-adjoint operator.

(d) $T \circ T^*$ is not diagonalizable.

2 Subjective Questions (24 marks)

1. Consider a matrix

$B = \begin{pmatrix} 1 & -3 & 3 \\ 3 & -5 & 3 \\ 6 & -6 & 4 \end{pmatrix}.$

(i) Find the inverse of the matrix $B$. [4 marks]

(ii) If $B$ is diagonalizable, then find a diagonal matrix $D$ and an invertible matrix $P$ such that $B = PDP^{-1}$. [6 marks]

(iii) What are the algebraic and geometric multiplicities of eigenvalues of $B^{-1}$? [2 marks]

1. Consider a matrix

$A = \begin{bmatrix} 2 & -i & i \\ i & 2 & -i \\ -i & i & 2 \end{bmatrix}.$

(a) Find the left eigenvector for at least one eigenvalue of $A$. [3 marks]

(b) Find the left null space of $A$. [3 marks]

1. Find the least squares solution to the system $Ax = b$, where

$A = \begin{bmatrix} 0 & 1 \\ 1 & 1 \\ 2 & 1 \end{bmatrix} \text{ and } b = \begin{bmatrix} 6 \\ 1 \\ -1 \end{bmatrix}.$

[6 marks]