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- Show that if v is an eigenvector of an n × n matrix A and v corresponds to a nonzero eigenvalue of A, then v is in Co l A. [Hint: Use the definition of an eigenvector.]Suppose V is finite-dimensional and S;T 2 L.V /. Prove that ST isinvertible if and only if both S and T are invertibleCompute the pseudoinverse A+ of A in the given exercise.
- a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish the same result without direct evaluation of the Wronskian.For what values of a does the matrix A=[01a1] have the characteristics below? a A has eigenvalue of multiplicity 2. b A has 1 and 2 as eigenvalues. c A has real eigenvalues.Identify which of the following functions are eigenfunctions of the operator d2/dx2: (i) cos(kx); (ii) eikx, (iii) kx, (iv) e-ax^2. Give the corresponding eigenvalue where appropriate.