• Attempt any five questions.
• All questions carry equal marks.
Define subspace of a vector space. Show that the quotient space of finitely generated space is finitely generated.
State and prove Rank nullity theorem.
Let f be linear functional on R³. Show that there exists a vector a in R³ such that f(r)=r·a.
Define Characteristic polynomials. Show that all eigen values of A*A are real and it is unitary similar to diagonal matrix.
State and prove Cayley Hamilton theorem.
Let T be a linear transformation on a vector space V of dimension n. Suppose that T has distinct eigen values. Then show that T is diagonalizable.
Define adjoint of Linear Transformation. Show that the given transformation T is a linear transformation.
Let W be a subspace of finite dimensional inner product space V and x∈V. Given the inequality condition, show that x∈W⊥.
Show that there is no proper open subspace of an inner product space.
Define Canonical form of linear transformation. Show that every m×n matrix is equivalent to unique matrix in one of canonical form.
Show that all eigen values of hermitian matrix are all real.
Define Jordan canonical form. Reduce the matrix [[1,0,0],[1,1,0],[0,0,3]] into Jordan canonical forms.
Show that a bilinear form on V is a product of linear functional iff it is of rank 1.
State and prove Gram Schmidt orthogonalization process.
Write short notes on:
i) Symmetric bilinear forms
ii) Group preserving bilinear forms
iii) Inner product space.
