1. Let X = =R2 be a two dimensional real vector space and let A be the matrix Define a mapping T : XX by Tx (6 3) . = Ax for x X. Suppose that X is endowed with X1 the 1-norm, that is ||x||1 := |x1|+|x2| for x = Є X. Find ||T||. X2

3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

5. Find the orthogonal projection of b = (1, –1,2) onto the left null space of the matrix A = |3

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= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).