• Attempt any five questions.
• All questions carry equal marks.
Show that u₁+u₂+u₃ is not a direct sum for given subspaces of F³.
Let F be a field of complex numbers and let T: F³→F³ be defined. Verify that T is linear transformation.
Suppose V is finite dimensional and U is subset to V. Show that U={0} if and only if U⁰=V'.
Write short note on:
i) Characteristic values and characteristic vectors
ii) Algebraic multiplicity and Geometric multiplicity.
Let A be a 4×4 real matrix. Show that the characteristic polynomial for A is x²(x-1)² and that it is also the minimal polynomial.
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.
Find the eigen values and eigen vectors of the matrix [[6,-2,2],[-2,3,-1],[2,-1,3]].
Explain invariant subspace with suitable examples.
Find Jordan canonical form of the matrix [[1,1,2],[1,2,1],[0,1,3]].
If T in A(V) has all its characteristic roots in F, then there exists a basis of V such that matrix representation of T is triangular. Prove it.
Find all Skew-symmetric bilinear forms on R³.
Find all bilinear forms on the space n×1 matrices over R which are invariant under O(n,R).
Write short note on the following (any two): a) Quotient spaces b) Direct sum decomposition c) Self adjoint d) Jordan blocks.
