CY-401 (GS) – Introduction to Linear Algebra

Prove: Let V be a vector space and W be a subspace of V. Then the map η:V→V/W defined by η(x)=x+W, x∈V, is a linear transformation.

Prove that, if V₀ is a subspace of a vector space V, then there exists a subspace V₁ of V such that V=V₀+V₁ and V₀∩V₁={0}.

Find two linear operators T and U on R² such that TU=0 but UT≠0.

Find the characteristic polynomial and the minimal polynomial for the matrix B=[[2,1,0],[0,2,0],[0,0,3]].

Prove that T is diagonalizable for the given matrix A=[[5,-6,-6],[-1,4,2],[3,-6,-4]]. Find the diagonalizable matrix P such that PAP⁻¹ is diagonal.

Given matrix B=[[5,2],[3,4]], use the Cayley-Hamilton theorem to calculate B²-9B+14 and verify that it results in the zero matrix.

Find the eigenvalues and eigenvectors of the symmetric matrix D=[[5,2,-1],[2,3,2],[-1,2,5]].

What are the essential properties of inner product spaces? Provide an example of a vector space that satisfies these properties.

Consider matrix A=[[5,4,2],[0,1,0],[0,0,3]]. Find the Jordan canonical form and determine the corresponding Jordan basis.

Show that the set of all continuous functions on [0,1] equipped with inner product ⟨f,g⟩=∫₀¹ f(x)g(x)dx forms an inner product space.

Let V be the space of n×n matrices over F. Define T(B)=AB-BA. Prove that if A is a nilpotent matrix, then T is a nilpotent operator.

Let C be a 5×5 matrix with characteristic polynomial (t-1)(t-2)²(t-3)². Determine the possible Jordan canonical forms of C.

Let T:C³→C³ be a linear transformation with matrix representation Λ=[[2,-i,0],[i,4,-3i],[0,3i,1]]. Determine whether T is Hermitian.

Suppose T:C²→C² is a linear transformation with matrix representation A. Determine whether T is unitary or not.

Write short note on any two: a) Direct sum decompositions b) Unitary and normal linear transformation c) Annihilator of a subspace.