a) Define subspace of a vector space. Show that the quotient space of finitely generated space is finitely generated. (Unit 1)

b) State and prove Rank nullity theorem. (Unit 1)

a) Let $f$ be linear functional on $R^{3}$. Show that $\exists\overline{a}\in R^{3}$ such that $f(\overline{r})=\overline{r}\cdot\overline{a}$. (Unit 1)

b) Define Characteristic polynomials. Show that all eigen values of $A^{*} A$ are real and it is unitary similar to diagonal matrix. (Unit 2)

a) State and prove Cayley Hamilton theorem. (Unit 2)

b) Let T be a linear transformation on a vector space V of dimension n. Suppose that T has distinct eigen values. Then show that T is diagonalizable. (Unit 2)

a) Define adjoint of Linear Transformation. The linear transformation $T:R^{3}\rightarrow R^{4}$ defined by $T(x,y,z)=(x-2y+z, 2x+y+2z, 3x-y+z)$ then show that T is a linear transformation. (Unit 3)

b) Let W be a subspace of finite dimensional inner product space V and $x\in V$ such that $+\le$ for all $y\in W$ then show that $x\in W^{\perp}?$ (Unit 3)

a) Show that there is no proper open subspace of an inner product space. (Unit 3)

b) Define Canonical form of linear transformation. Show that every $m\times n$ matrix is equivalent to unique matrix in one of canonical form. (Unit 4)

a) Show that all eigen values of hermitian matrix are all real. (Unit 5)

b) Define Jordan canonical form. Reduce the matrix
$\begin{bmatrix}1&0&0\\ 1&1&0\\ 0&0&3\end{bmatrix}$
Into Jordan canonical forms. (Unit 4)

a) Show that a bilinear form on V is a product of linear functional iff it is of rank 1. (Unit 5)

b) State and prove Gram Schmidt orthogonalization process. (Unit 3)

Write short notes on

i) Symmetric bilinear forms (Unit 5)

ii) Group preserving bilinear forms (Unit 5)

iii) Inner product space (Unit 3)