• Attempt any five questions.
• All questions carry equal marks.
Suppose V is finite dimensional and U is subspace of V. Show that U=V if and only if U⁰={0}.
Show that the mapping T: V₃(R)→V₂(R) defined as T(a₁,a₂,a₃)=(3a₁-2a₂+a₃, a₁-3a₂-2a₃) is a linear transformation.
Suppose V is finite dimensional and U is a subspace of V. Then prove that dim(V/U)=dim V - dim U.
Write short note on:
i) Cayley-Hamilton theorem
ii) Annihilating Polynomials.
Let a, b, c be elements of a field F, and let A be the 3×3 matrix over F. Prove that the characteristic polynomial for A is x³-ax²-bx-c and that it is also the minimal polynomial for A.
Let A be an n×n triangular matrix over the field F. Prove that characteristic values of A are the diagonal entries of A.
Diagonalize the matrix [[1,0,-1],[1,2,1],[2,2,3]].
Show that V₂(R) is an inner product space defined by (α,β)=3a₁b₁+2a₂b₂ for all α=(a₁,a₂), β=(b₁,b₂) in V₂(R).
Explain Jordan blocks with suitable examples.
Show that two nilpotent linear transformations are similar if and only if they have same invariants.
Find a basis for the space of all skew-symmetric linear forms on Rⁿ.
Find all bilinear forms on the space of n×1 matrices over C which are invariant under O(n,C).
Write short note on following (any two): a) Dual Spaces b) Primary decomposition theorem c) Bilinear forms d) Inner product spaces.
