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Q3
Let T : R 2 R R be a function. The following three statements would be true if T were a linear transformation:
(i) {Tx|xeR2} is a subspace of R.
(ii) {x e R2 Tx = 0} is a subspace of R2.
(iii) lf x E R2 and CER are such that Tx = C, then T-1(c)={x}+{yeR2Ty=0
Which of these statements are true and which are false for the nonlinear function T:R2- R defined by Tx=+-1 for every
X = (X1,X2) E R2? Prove your answers
Q5(a)
For a polynomial with coefficients in C. and A E Mnxn(C), we define
If p is a polynomial such that p(A) = 0, then we say p annihilates A
Show that if p is a polynomial, and X is an eigenvector of A with eigenvalue l then X is an eigenvector of p(A) with eigenvalue p(X).
(b)
Let p be a polynomial that annihilates A. Must every eigenvalue of A be a root of p(z)? Prove your answer.
(c)
Prove that there exists a non-constant polynomial p of degree at most n2 that annihilates A. Hint: Use the fact that dim Mnxn(C) = n2. What
can you say about the n2 + 1 matrices I, A, A2 ...A
Q6(a)
Let W be a subspace of an n-dimensional vector space V over C. and let T:V-V be a linear transformation.
Suppose W is invariant under T, and that V - ker T + W Prove that W = im T
(b)
Prove that W is invariant under T if and only if W is invariant under T-XI - for any AEC
(c)
Suppose T is diagonalizable and let be the distinct eigenvalues of T. Prove that

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