a) Show that $u_{1}+u_{2}+u_{3}$ is not a direct sum if
$u_{1}=\{(x,y,0)\in F^{3}where~x,y\in F\}$
$u_{2}=\{(0,0,z)\in F^{3}wherez\in F\}$
$u_{3}=\{(0,y;y)\in F^{3}where,y\in F\}$ (Unit 1)

b) Let F be a field of complex numbers and let $T:F^{3}\rightarrow F^{3}$ defined by T(x1 3) = ( -x2 + 2x3.2x1 + x2-x3, $x_{1}-2x_{2})$. Verify that T is linear transformation. (Unit 1)

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

b) Write short note on following:

i) Characteristic values and characteristic vectors (Unit 2)

ii) Algebraic multiplicity and Geometric multiplicity (Unit 2)

a) Let A be $4\times4$ 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 it is also the minimal polynomial. (Unit 2)

b) Let T be a linear operator on the n - dimensional vector space V and suppose that T has n - distinct characteristic values. Prove that T is diagonalizable. (Unit 2)

Find the eigen values and eigen vectors of
$A=\begin{bmatrix}6&-2&2\\ -2&3&-1\\ 2&-1&3\end{bmatrix}$ (Unit 3)

a) Explain invariant subspace with suitable examples. (Unit 3)

b) Find Jordon canonical form of the matrix
$A=\begin{bmatrix}1&1&2\\ 1&2&1\\ 0&1&3\end{bmatrix}.$ (Unit 4)

If $T\in A(V)$ has all it's characteristic roots in F, then there exists a basis of V such that matrix representation of T is triangular. Prove it. (Unit 4)

a) Find all Skew - symmetric bilinear forms on $R^{3}$. (Unit 5)

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

Write short note on the following (any two):

a) Quotient spaces (Unit 1)

b) Direct sum decomposition (Unit 2)

c) Self adjoint (Unit 5)

d) Jordan blocks (Unit 4)