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Honours Algebra 2 (2) (a) Let S(2) = (~e1, ~e2) be the standard basis of R2 and let B = (~v1 = 3~e1+2~e2, ~v2 = 2~e1 ~e2). Show that B is a basis of R2 . Now suppose that a linear mapping f : R2 → R2 is represented with respect to S(2) by the matrix A =  6 9 4 6  Find the matrix B that represents f with respect to B. Write down an explicit equation that expresses the relationship between A and B. [7 marks] (b) In each of the following cases state whether the given formula defines an inner product on R2 . For any that are not inner products, give a counterexample; for any that are inner products, a proof is not required. In each formula ~x = (x1, x2)T and ~y = (y1, y2)T. (i) (~x, ~y) = x1y1 + 2x1y2 + 2x2y1 + 3x2y2. (ii) (~x, ~y) = x21y21 + 5x2y2. (iii) (~x, ~y) = x1y1 + x1y2 + x2y1 + 3x2y2. [5 marks] (c) Let A ∈ Mat(4; C) be the following matrix: A = ???? 2 1 3 2 4 4 4 4 4 4 4 4 6 5 7 6???? There exists a matrix P ∈ Mat(4; C) such that P 1AP = ???? 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4???? [You do not need to prove this.] Find P. [8 marks] [Please turn over] MATH10069 Honours Algebra 3 (3) (a) Define what it means for a ring to be an integral domain. State and prove the cancellation theorem for integral domains. Prove that Z[X] is an integral domain. [5 marks] (b) Let V be a finite dimensional complex inner product space. Define the norm on V and state and prove the Cauchy-Schwarz inequality. [8 marks] (c) Let A be a matrix with entries in a field F. Define the determinant det(A), explaining any terminology you use. Prove that det(AT ) = det(A). Define χA(x) ∈ F[x], the characteristic polynomial of A and show that the eigenvalues of A in F are exactly the roots of the polynomial χA(x). [7 marks] [Please turn over] MATH10069 Honours Algebra 4 (4) (a) Let f : R → S be a ring homomorphism. Define the kernel and image of f and prove that the image of f is a subring of S. State the First Isomorphism Theorem for Rings. [8 marks] (b) Let A ∈ Mat(3; F2). Show that the mapping fA : F2[X] → Mat(3; F2), defined by p0 + p1X + p2X2 + · · · + pnXn 7→ p0I + p1A + p2A2 + · · · + pnAn for any p0 + p1X + p2X2 + · · · + pnXn ∈ F2[X], is a ring homomorphism. Using the First Isomorphism Theorem or otherwise, prove that the image of fA is a field with 8 elements when A = ?? 0 0 1 1 0 0 0 1 1 ?? . [12 marks]

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