a) Let $U_{1}$, $U_{2}$ and $U_{3}$ be subspaces of a vector space V. Prove that
$U_{1}\cap(U_{2}+(U_{1}\cap U_{3}))=(U_{1}\cap U_{2})+(U_{1}\cap U_{3})$ (Unit 1)

b) Let V be a vector space. Show that $dim(V)=n\ge1$ if and only if there exist one-dimensional subspaces $U_{1}...U_{n}$ such that $V=U_{1}\oplus...\oplus U_{n}$ (Unit 1)

a) Let V be a vector space over a field F and $T:V\rightarrow V$ be a linear transformation. Define the annihilator of a subspace W of V as
$W^{0}=\{f\in V^{*}|f(w)=0 \text{ for all } w\in W\}$,
Where $V*$ is the dual space of V. Prove that $W^{0}$ is a subspace of $V*$ and determine its dimension in terms of the dimension of V and W. (Unit 1)

b) Let A be the $4\times4$ real matrix
$A=\begin{bmatrix}1&1&0&0\\ -1&-1&0&0\\ -2&-2&2&1\\ 1&1&-1&0\end{bmatrix}$
Show that the characteristic polynomial for A is $x^{2}(x-1)^{2}$ and that is also the minimal polynomial. (Unit 2)

a) Let W be an invariant subspace for linear operator T. Prove that the minimal Polynomial for the restriction operator $T_{w}$ divides the minimal polynomial for T, without referring to matrices. (Unit 2)

b) For the matrix A:
$A=\begin{bmatrix}3&1\\ 2&2\end{bmatrix}$
Verify Cayley-Hamilton theorem by finding its characteristic polynomial and substituting the matrix into it. (Unit 2)

a) Consider the matrix A:
$A=\begin{bmatrix}6&1&0\\ 0&6&1\\ 0&0&6\end{bmatrix}$
Find the Jordan canonical form of the matrix A and the corresponding Jordan basis. (Unit 4)

b) Let V be an inner product space, and let $\alpha$ and $\beta$ be vectors in V. Show that $\alpha=\beta$ if and only if $(\alpha|\gamma)=(\beta|\gamma)$ for every $\gamma$ in V. (Unit 3)

a) Consider a matrix A given by:
$A=\begin{bmatrix}3&1\\ 1&3\end{bmatrix}$
i) Find the eigen values and corresponding eigen vectors of A.
ii) Determine the invariant subspaces corresponding to each distinct eigen value. (Unit 3)

b) Given that the set $\{(1,0,0),(0,2,1),(2,0,1)\}$ is a basis of $\Re^{3}$. If $T:\Re^{3}\rightarrow\Re^{3}$ is a linear transformation such that $T(1,0,0)=(0,0,1),$ $T(0,2,1)=(1,2,0)$ and $T(2,0,1)=(1,1,1)$. Find $T(2,-3,4)$. (Unit 1)

a) If V is the space of all polynomials of degree less than or equal to n over a field F, prove that the differentiation operator on V is nilpotent. (Unit 4)

b) Consider a $2\times2$ matrix G with characteristic polynomial $(t+1)^{2}(t-2)$ and minimal polynomial $(t+1)(t-2)$. Determine the primary decomposition of $R^{2}$ with respect to G. (Unit 4)

a) Let A, $B\in O(n)$, the orthogonal group of real $n\times n$ matrices. Define the bilinear form as:
$B(A,B)=tr(A^{T}B)$
Verify if B preserves the group structure of $O(n)$. (Unit 5)

b) Write a detail note on skew symmetric bilinear forms. (Unit 5)

Write short note on any two:

a) Cayley-Hamilton theorem (Unit 2)

b) Primary decomposition theorem (Unit 4)

c) Invariant subspaces (Unit 3)