a) Prove: Let V be a vector space and W be a subspace of V. Then the map $\eta:V\rightarrow V/W$ defined by $\eta(x)=x+W,x\in V$, is a linear transformation. (Unit 1)

b) Prove that, if $V_{0}$ is a subspace of a vector space V, then there exists a subspace $V_{1}$ of V such that $V=V_{0}+V_{1}$ and $V_{0}\cap V_{1}=\{0\}$. (Unit 1)

a) Find two linear operators T and U on $R^{2}$ such that $TU=0$ but $UT\ne0.$ (Unit 1)

b) Find the characteristic polynomial and the minimal polynomial for the matrix:
$B=\begin{bmatrix}2&1&0\\ 0&2&0\\ 0&0&3\end{bmatrix}$ (Unit 2)

a) If T be the linear operator on R3 which is represented in the standard basis by the matrix
$A=\begin{bmatrix}5&-6&-6\\ -1&4&2\\ 3&-6&-4\end{bmatrix}$
Prove that T is diagonalizable. Find the diagonalizable matrix P that $PAP^{-1}$ is diagonal. (Unit 2)

b) Given a $2\times2$ matrix:
$B=\begin{bmatrix}5&2\\ 3&4\end{bmatrix}$
Use the Cayley-Hamilton theorem to calculate $B^{2}-9B+14$ and verify that it results in the zero matrix. (Unit 2)

a) Let D be a symmetric matrix given by:
$D=\begin{bmatrix}5&2&-1\\ 2&3&2\\ -1&2&5\end{bmatrix}$
Find the eigenvalues and eigenvectors of D. (Unit 3)

b) What are the essential properties of inner product spaces? Provide an example of a vector space that satisfies these properties. (Unit 3)

a) Consider the matrix A given by:
$A=\begin{bmatrix}5&4&2\\ 0&1&0\\ 0&0&3\end{bmatrix}$
i) Find the Jordan canonical form of the matrix A.
ii) Determine the corresponding Jordan basis. (Unit 4)

b) Show that the set of all continuous functions on the interval [0, 1] equipped with the inner product $\langle f,g\rangle=\int_{0}^{1}f(x)g(x)dx$ forms an inner product space. Justify that this definition meets the properties of an inner product. (Unit 3)

a) Let V be the space of $n\times n$ matrices over a field F, and let A be a fixed $n\times n$ matrix over F. Define a linear operator T on V by $T(B)=AB-BA$ Prove that if A is a nilpotent matrix, then T is a nilpotent operator. (Unit 4)

b) Let C be a $5\times5$ matrix with characteristic polynomial $(t-1)(t-2)^{2}(t-3)^{2}.$ Determine the possible Jordan canonical forms of C. (Unit 4)

a) Let T $:C^{3}\rightarrow C^{3}$ be a linear transformation with the matrix representation:
$\Lambda=\begin{bmatrix}2&-i&0\\ i&4&-3i\\ 0&3i&1\end{bmatrix}$
Determine whether T is Hermitian. (Unit 5)

b) Suppose T: $C^{2}\rightarrow C^{2}$ is a linear transformation with the matrix representation:
$A=\begin{pmatrix}1/\sqrt{2}&-i/\sqrt{2}\\ i/\sqrt{2}&1/\sqrt{2}\end{pmatrix}$
Determine whether T is unitary or not. (Unit 5)

Write short note on any two:

a) Direct sum decompositions (Unit 2)

b) Unitary and normal linear transformation (Unit 5)

c) Annihilator of a subspace (Unit 1)