• Attempt any five questions.
• All questions carry equal marks.
Let U₁, U₂ and U₃ be subspaces of a vector space V. Prove that U₁∩(U₂+(U₁∩U₃))=(U₁∩U₂)+(U₁∩U₃).
Let V be a vector space. Show that dim(V)=n≥1 if and only if there exist one-dimensional subspaces U₁...Uₙ such that V=U₁⊕...⊕Uₙ.
Define the annihilator of a subspace W of V. Prove that W⁰ is a subspace of V* and determine its dimension in terms of the dimension of V and W.
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 W be an invariant subspace for linear operator T. Prove that the minimal polynomial for the restriction operator T_w divides the minimal polynomial for T.
For the matrix A=[[3,1],[2,2]], verify Cayley-Hamilton theorem by finding its characteristic polynomial and substituting the matrix into it.
Consider the matrix A=[[6,1,0],[0,6,1],[0,0,6]]. Find the Jordan canonical form of the matrix A and the corresponding Jordan basis.
Let V be an inner product space, and let α and β be vectors in V. Show that α=β if and only if (α|γ)=(β|γ) for every γ in V.
Consider matrix A=[[3,1],[1,3]]. Find the eigenvalues and corresponding eigenvectors of A. Determine the invariant subspaces corresponding to each distinct eigenvalue.
Given basis and linear transformation values, find T(2,-3,4).
If V is the space of all polynomials of degree less than or equal to n over a field F, prove that the differentiation operator on V is nilpotent.
Consider a 2×2 matrix G with characteristic polynomial (t+1)²(t-2) and minimal polynomial (t+1)(t-2). Determine the primary decomposition of R² with respect to G.
Let A, B in O(n), the orthogonal group of real n×n matrices. Define the bilinear form as B(A,B)=tr(AᵀB). Verify if B preserves the group structure of O(n).
Write a detail note on skew symmetric bilinear forms.
Write short note on any two: a) Cayley-Hamilton theorem b) Primary decomposition theorem c) Invariant subspaces.
