# matrix algebra

| April 27, 2015

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BSc/MSci/MSc EXAMINATION 2014 For internal students of Royal Holloway DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

For internal students of Royal Holloway DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

BSc/MSci/MSc EXAMINATION 2014 For internal students of Royal Holloway DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

For internal students of Royal Holloway DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

DO NOT TURN OVER UNTIL TOLD TO BEGIN MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

MT1820: MATRIX ALGEBRA MT1820R: MATRIX ALGEBRA – PAPER FOR RESIT CANDIDATES Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

Time Allowed: TWO hours All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R

All questions may be attempted but credit will only be given for the best four answers. A borderline first class mark would normally require answers to the equivalent of three complete questions. MT and Type A Calculators are permitted. °c Royal Holloway, University of London 2014 Page 1 of 4 2013–14 Important Copyright Notice This exam paper has been made available in electronic form strictly for the educational benefit of current Royal Holloway students on the course of study in question. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 2 of 4 MT1820 1. Let A, B, C, D be points in R 3 with position vectors a =   1 2 −2   , b =   2 1 4   , c =   −1 3 2   , d =   3 3 2   respectively. (a) Calculate a · b and (b − a) × (c − a). (b) Calculate the area of the triangle ABC. (c) Find an equation of the plane through A, B, C in the form v · n = d. (d) Do the position vectors of the points of the plane through A, B, and C form a subspace of R 3 ? Justify your answer. (e) Do the position vectors of the points of the plane through A, B, D form a subspace of R 3 ? Justify your answer. (f) Find a plane P parallel to the plane through A, B, C such that the position vectors of P form a subspace of R 3 . 2. Which of the following statements are true? Justify your answers. (a) Let (V, +, ·) be a vector space over a field F. If {a, b} forms a basis of V then {a, a+b} forms a basis of V . (b) Let A and B be invertible n × n matrices. Then (AB) −1 = B−1A−1 . (c) For all integers n ≥ 2 and for all n × n matrices A and B, we have det(A + B) = det(A) + det(B). (d) For every linear map f : R s → R t , there exists an t × s matrix that represents f. (e) If f is a linear map then for all vectors a, we have f(−a) = −f(a). NEXT PAGE No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. MT1820 Page 3 of 4 2013–14 3. (a) Define a permutation of {1, . . . , n}. (b) Let σ = Ã 1 2 3 4 5 6 7 8 4 1 8 5 7 3 2 6 ! . (i) Write σ as a product of disjoint cycles. (ii) Write σ as a product of transposition. (iii) Can σ be written as a product of 1000 transpositions? Justify your answer. (iv) Can σ be written as a product of 1001 transpostions? Justify your answer. (c) Define what it means for a matrix to be invertible. (d) Prove that the inverse of an invertible matrix is unique. (e) Which of the following matrices are invertible? Justify your answer. (i) A = Ã 1 3 3 4 3 8 ! , B =   1 3 3 1 3 3 4 3 8  , C =   1 0 0 0 3 0 0 0 2  . 4. (a) Find the determinant and the inverse of the following matrix A using elementary row operations and verify that you have indeed found the inverse. A =   0 2 1 2 2 2 3 1 0   . (b) Let f : R 2 → R 2 be the linear map that satisfies f ÃÃ 1 0 !! = Ã 2 3 ! and f ÃÃ 1 1 !! = Ã 2 2 ! (i) Find the matrix B that represents f, that is, the matrix B that satisfies Bx = f(x) for all x ∈ R 2 . (ii) Calculate f ÃÃ 4 3 !!. (iii) Is f bijective? Justify your answer. NEXT PAGE No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. 2013–14 Page 4 of 4 MT1820 5. (a) Let (V, +, ·) be a vector space over a field F. What does it mean for a family of vectors s1, . . . , sn of V to be (i) linearly independent? (ii) spanning for V ? (iii) a basis of V ? (b) Consider (R 4 , +, ·) over R and the following vectors: a =   1 0 2 3   , b =   1 −1 1 0   , c =   2 1 1 2   , d =   3 0 2 s   . (i) Does {a, b, c} form a basis of R 4 ? (ii) For which value(s) of s is {a, b, c, d} linearly independent? (iii) For which value(s) of s is {a, b, c, d} spanning for R 4 ? (iv) If s = 2 what is the dimension of span({b, c, d})? Justify your answer. (v) If s = 1 what is the dimension of span({a, b, d})? Justify your answer. END S. Gerke No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions. No further copying, distribution or publication of this exam paper is permitted. By printing or downloading this exam paper, you are consenting to these restrictions.

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