This paper is not to be removed from the Examination Halls

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1 ~~MT1173 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON MT1173 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences and Access Route Algebra Monday, 11 May 015 : 10:00 to 13:00 Candidates should answer all FIVE questions. All questions carry equal marks. Calculators may not be used for this paper. University of London 015 UL15/0317 Page 1 of 6 D1

2 1. Consider the following system of linear equations for constants a and b and unknowns x,y,z,w. ax+y +z +w = 1 x+ay z 4w = 1 x+z +w = b (a) Express the system in matrix form as Ax = b. (b) Show that A has rank if and only if a =. (c) If a =, show that the equations are consistent if and only if b = 1. (d) If a = and b = 1, find the general solution of the system using Gaussian elimination. Express your solution in vector form. (e) State a condition that the ranks of the coefficient matrix A and the augmented matrix (A b) must satisfy for the system of equations Ax = b to be consistent. Deduce that if a the system of equations Ax = b is consistent for all b R. When the system is consistent, does it have a unique solution? (f) Consider the set of vectors S = {v 1,v,...,v k } in a vector space V. Define what it means to say that (i) S is linearly independent and (ii) S spans V. (g) For the matrix A you gave in part (a), answer each of the following questions, briefly justifying your answers. (i) For what values of a, if any, are the column vectors of A linearly independent? (ii) For what values of a, if any, are the row vectors of A linearly independent? (iii) For what values of a, if any, do the column vectors of A span R 3? (iv) For what values of a, if any, do the row vectors of A span R 4? UL15/0317 Page of 6 D1

3 . (a) A system of linear equations Ax = b is known to have the following general solution: x = 0 +s 1 0 +t 0 1 s,t R The first column of A is the vector c 1 = If b = 3 5, find the matrix A (b) Let l 1 be the line in R 3 with equation x y z = t 1, t R 3 and let l be the line with equation x y = 7 1 +t 1 1, t R. z 4 (i) Write down an expression for the set S 1 of position vectors of points on the line l 1. Prove that S 1 is a subspace of R 3. (ii) If S is the set of position vectors of points on the line l, show that S is not a subspace of R 3. (iii) Show that the two lines l 1 and l intersect and find the point of intersection. (iv) Find a Cartesian equation of the plane containing both lines. Is the set of position vectors of points on this plane a subspace of R 3? If it is a subspace, write down a basis. If not, justify your answer. UL15/0317 Page 3 of 6 D1

4 3. (a) What does it mean to say that two n n matrices A and B are similar? What does it mean to diagonalise an n n matrix A? Prove the following statement. If two diagonalisable n n matrices A and B have the same eigenvalues, λ 1,...λ n, then they are similar. (b) Consider the matrices A = ( ) and B = ( ) (i) Diagonalise A. (ii) Show that B has the same eigenvalues as A and diagonalise the matrix B. (iii) Find an invertible matrix P such that B = P 1 AP. Consider P as the transition matrix from standard coordinates in R to coordinates in a basis S = {s 1,s }. Write down the basis S. (iv) Let T : R R be the linear transformation defined by T(x) = Ax in standard coordinates. If S is the basis in part (iii), explain why B represents the same linear transformation T with respect to the basis S; that is, show that B[x] S = [T(x)] S. [ ] 3 Verify this for the vector q R with [q] S = by finding T(q) using S the matrix A and then using the matrix B. UL15/0317 Page 4 of 6 D1

5 4. (a) Let T be a linear transformation defined by T(x) = Ax where A is the matrix A = and let b = (i) Put the matrix A into reduced row echelon form. (ii) Find a basis of the null space, N(T), of T. (iii) Find a basis of the range, R(T), of T. (iv) State the rank-nullity (dimension) theorem for linear transformations, carefully defining each term, and verify it for the linear transformation T. (v) Show that b R(T). Find all vectors x for which T(x) = b. (b) A market for a commodity is modelled by the supply and demand functions, q S and q D, respectively, of the market price p, where: q S (p) = p and q D (p) = 3 p. The price change p t p t 1 depends on the excess demand in the previous two periods according to the equation p t p t 1 = 3 8 ( ) q D (p t 1 ) q S (p t 1 ) 1 ( ) q D (p t ) q S (p t ). 16 Using the above information, show that p t satisfies the second-order difference equation p t 1 4 p t p t = Given that p 0 = 5 and p 1 = 1, find p t. Describe in words the behaviour of p t as t increases. UL15/0317 Page 5 of 6 D1

6 5. Consider the vectors v 1 = 1 0, v = 1 1 1, v 3 = 3. (a) Show that the set B = {v 1,v,v 3 } is a basis of R 3. Write down the transition matrix P from coordinates in the B basis to standard coordinates. Find P 1. If [u] B = 6 1, find u in standard coordinates. B If w = 3 in standard coordinates, find [w] B. 1 (b) Find a matrix A for which v 1, v and v 3 are eigenvectors corresponding to eigenvalues λ 1 = 1, λ = and λ 3 = 3, respectively. Verify that your matrix A has the required property for the eigenvector v 3. (c) For the matrix A in part (b), sequences x t, y t, z t are known to satisfy the system of difference equations x t+1 = Ax t, where x t = x t y t and t 0, z t with x 0 = w, where w is the vector given in part (a). Solve this system to find expressions for each of the sequences x t, y t, z t. Write down the vector x 1. END OF PAPER UL15/0317 Page 6 of 6 D1