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LANCASTER UNIVERSITY
2012 EXAMINATIONS
PART II (Second year) MATHEMATICS & STATISTICS 2 hours Math 220: Linear Algebra
You should answer ALL Section A questions and THREE Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40 SECTION A
A1. Determine whether or not the vectors (1, โ 1 , 3), (2, 1 , 4) and (1, โ 3 , 2) are linearly independent in R^3. [5] A2. Consider the system of linear equations 3 x + y โ 3 z = 5 4 x + 2 y โ 2 z = 8 x + y + z = 3
(i) Write down the augmented matrix for the above system. [1] (ii) Find the reduced echelon form for the matrix from (i). [3] (iii) Use (ii) to find the solution to the system. [1] (iii) Interpret your solution geometrically. [1] A3. Let S = {(2, 1 , โ1), (3, 1 , 4), (โ 1 , 0 , 5), (1, 1 , โ6)}. Find the dimension of the subspace of R^3 spanned by S. [6]
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SECTION A continued
A4. Let T : R^3 โ R^2 be given by T ((x, y, z)) = (x + 2y + z, x โ y + z). (i) Prove that T is a linear transformation. [3] (ii) Find a basis for the kernel of T , ker T. [3] (iii) Find a basis for the image of T , im T. [3] (iv) Is T (a) onto, (b) one-one? Give brief reasons. [2] (v) Write down the matrix of T with respect to the standard bases in the domain and codomain. [1] (vi) Write down the transition matrix from the basis (1, 0 , 0), (1, 1 , 0), (1, 1 , 1) to the standard basis in R^3. [1] (vii) Write down the transition matrix from the basis (1, 1), (1, โ1) to the standard basis in R^2. [1] (viii) Use (v), (vi) and (vii) to calculate the matrix of T with respect to the basis (1, 0 , 0), (1, 1 , 0), (1, 1 , 1) in the domain and the basis (1, 1), (1, โ1) in the codomain. [4] A5. By calculating the ranks of two matrices, show that the system of equations below has infinitely many solutions: โx + 4y + 2z = 1 3 x + 2y + 4z = 7 2 x โ y + z = 3. [5] A6. Let A =
(i) State the Cayley-Hamilton Theorem. [2] (ii) Find the characteristic polynomial of A. [3] (iii) Use the Cayley-Hamilton Theorem to find Aโ^1. [5]
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SECTION B continued
B3. (^) (i) Explain what is meant by the terms (a) inner product, and (b) Euclidean space. [5] (ii) Let V = P 2 be the vector space of all polynomials of degree less than or equal to two. (a) Prove that ใp(x), q(x)ใ = โซ^ โ^11 p(x)q(x)dx defines an inner product on P 2. [6] (b) Apply the algorithm
f 1 = e 1 , fi = ei โ โ^ iโ^1 k=
ใei, fkใ โ fk โ^2 fk^ for 2^ โค^ i^ โค^ n (with the inner product defined in (a)) to the basis 1, x, x^2 to produce an orthogonal basis for P 2. [9] B4. Let A =
(i) Determine the characteristic equation of A, and hence find the eigenvalues of A. [5] (ii) Determine the minimal polynomial of A. [3] (iii) Write down the Jordan normal form J for A. [2] (iv) Find a Jordan basis for A. [10]
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