Introduction to Linear Algebra (CY-401) - Important Questions

Let W1 and W2 be subspaces of R^3 given by W1 = {(x, y, 0) : x, y in R} and W2 = {(0, 0, z) : z in R}. Determine whether R^3 = W1 + W2 is a direct sum, i.e., whether R^3 = W1 (+) W2.

The set {(1,0,0), (0,2,1), (2,0,1)} is a basis of R^3. If T: R^3 -> R^3 is a linear transformation such that T(1,0,0) = (0,0,1), T(0,2,1) = (1,2,0) and T(2,0,1) = (2,1,2), find T(x,y,z) for general (x,y,z) and find T(1,1,1).

Let T be the linear operator on R^3 represented in the standard basis by the matrix A = [[5,-6,-6],[-1,4,2],[3,-6,-4]]. Show that T is diagonalizable and find the diagonal matrix D and the transforming modal matrix P such that P^{-1}AP = D.

For the matrix A = [[3,1],[2,2]], verify the Cayley-Hamilton theorem by finding its characteristic polynomial and substituting the matrix into it to show that it satisfies its own characteristic equation.

Prove that a linear operator T on an n-dimensional vector space V having n distinct characteristic values / eigenvalues is diagonalizable.

Let A = [[1,1,0,0],[-1,-1,0,0],[-2,-2,2,1],[1,1,-1,0]]. Show that the characteristic polynomial for A is x^2(x-1)^2 and that it is also the minimal polynomial for A.

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

Show that the set of all continuous functions on the interval [0,1] equipped with the inner product <f,g> = integral_0^1 f(x)g(x)dx forms an inner product space by verifying the inner product axioms.

Define the adjoint of a linear transformation on a finite-dimensional inner product space. If T: R^3 -> R^3 is defined by T(x,y,z) = (x+2y-z, y+z, x+y-2z), find T* and its matrix in the standard orthonormal basis.

Consider the matrix A = [[6,1,0],[0,6,1],[0,0,6]]. Find the Jordan canonical form of the matrix A and determine the corresponding Jordan basis.

Find the Jordan canonical form of the matrix B = [[2,1,0],[0,2,0],[0,0,3]] and hence determine its minimal polynomial.

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 D on V is a nilpotent linear transformation and find its index of nilpotency.

Find a basis and the general form for the space of all skew-symmetric bilinear forms on R^3.

Find all bilinear forms on the space of n x 1 column vectors over R which are invariant under the orthogonal group O(n).

Write short notes on any two of the following: (i) Quotient spaces and canonical map V -> V/W (ii) Dual spaces and dual basis (iii) Annihilator of a subspace.