a) Suppose V is finite dimensional and U is subspace of V. Show that $U=V$ if and only if $U^{0}=\{0\}$. (Unit 1)

b) Show that the mapping $T:V_{3}(R)\rightarrow V_{3}$ (R) defined as $T(a_{1},a_{2},a_{3})=(3a_{1}-2a_{2}+a_{3},a_{1}-3a_{2}-2a_{3})$ is a linear transformation from $V_{3}$ (R) into $v_{2}$ (R). (Unit 1)

a) Suppose V is finite dimensional and U is a subspace of V. Then prove that $dim(\frac{V}{M})=dimV-dimU$ . (Unit 1)

b) Write short note on:

i) Cayley-Hamilton theorem (Unit 2)

ii) Annihilating Polynomials (Unit 2)

a) Let a, b, c be elements of a field F, and let A be the following $3\times3$ matrix over F, $A=\begin{bmatrix}0&0&c\\ 1&0&b\\ 0&1&a\end{bmatrix},$ prove that the characteristic polynomial for A is $x^{3}-ax^{2}-bx-c$ and that it is also the minimal polynomial for A. (Unit 2)

b) Let $n\times n$ triangular matrix over the field F. Prove that characteristic values of A are the diagonal entries of A. (i.c., the scalars $A_{ii}$). (Unit 2)

Diagonalize the following matrices

$\begin{bmatrix}1&0&-1\\ 1&2&1\\ 2&2&3\end{bmatrix}.$ (Unit 3)

a) Show that $V_{2}$ (R) is an inner product space defined by $(\alpha,\beta)=3a_{1}b_{1}+2a_{2}b_{2}\forall,\alpha=(a_{1},a_{2}),\beta=(b_{1},b_{2})\in V_{2}(R)$ (Unit 3)

b) Explain Jordan blocks with suitable examples. (Unit 4)

Show that "Two nilpotent linear transformations are similar if and only if they have same invariants". (Unit 4)

a) Find a basis for the space of all skew-symmetric linear forms on $R^{n}$ (Unit 5)

b) Find all bilinear forms on the space of $n\times1$ matrices over C which are invariant under $O(n,c)$. (Unit 5)

Write short note on following (any two):

a) Dual Spaces (Unit 1)

b) Primary decomposition theorem (Unit 4)

c) Bilinear forms (Unit 5)

d) Inner product spaces (Unit 3)