F07 - Matrices and Linear Spaces - Tutorial Set 3 (modified)

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 1 of 17 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Matrices and Linear Spaces (Tutorial Set 3) This tutorial set is for the following sections from the notes: §7 Row Space, Column Space and Null Space §8 Linear Transformations §9 Eigenvalues and Eigenvectors Basic Mastery Questions 1 For each of the following matrices, find the bases for the row space, c olumn space and null space. (a) 1 3 2 1 1 1         , (b) 1 2 1 3 5 1 13 23 7         , (c) 2 1 3 3 0 3 1 2 4 5 5 8         . State the rank and the nullity for each of these matrices. 2 Determine whether each of the following is a linear transformation. Justify your answers. (a) 2 2 1T :   . 1T x kx y y           , where k is a nonzero real constant. (b) 2 2 2T :   . 2T x yx x yy            . (c)   2 3 2,2T : Μ   , 3T a b a b c d c d            . (d) 4 2T : P  ,   2 2 4T 4ax bx c b ac    . (e) 3 4 5T :   ,  5T u 0 for all 3 u  . 3 It is given that the linear transformation 2 2 T :   is such that 2 11T 3 13           and 3 8T 4 11            . (i) Find 1T 7      . (ii) Find the 2 2 matrix A such that  T u Au for all 2 u  . www.KiasuExamPaper.com 707

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 2 of 17 4 The linear transformation 3 3 L :   can be represented by the matrix 2 1 2 1 2 1 2 3 2 4 a a a a         It is given the rank of L is 2. (i) State the nullity of L and find the value of a. (ii) Find a basis for its range space. 5 Find the eigenvalues and eigenvectors of the following matrices. (a) 2 4 5 3      , (b) 6 4 8 6 11 3 4 6 12           , (c) 0 2 1 1 3 1 2 4 1         . 6 Let 2 4 5 3     A . (i) Find an invertible matrix P and a diagonal matrix D such that 1 P AP D . (ii) Find 5A without using a graphic calculator. 7 [Removed] Practice Questions 8 The subspaces V and W of 3 are spanned by the sets 1 2 3 3 , 0 , 5 2 5 0                                 and 1 2 2 3 , 4 , 0 3 3 3                                  respectively. (i) Find the dimensions of V and of W. (ii) Given that x y V z         , obtain a linear relationship between x, y and z. (iii) Find a 1 3 matrix A such that  : W X AX 0 . (iv) Find a basis for the subspace V W . (1983 A Level / FM / Jun / P2) www.KiasuExamPaper.com 708

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 3 of 17 9 (a) Given that  a b c belongs to the row space of the matrix 3 2 1 2 2 1 1 2 7          , find a linear relation hat must be satisfied by a, b, c. (b) Given that P and Q are 3 3 matrices, (i) prove that the column space of PQ is a subspace of the column space of P, (ii) state a similar result concerning the row space of PQ, (iii) deduce that  rank PQ cannot exceed the smaller of  rank P and  rank Q . (1984 A Level / FM / Jun / P1) 10 The elements of the matrices A and B are given by 11 12 13 21 22 23 31 32 33 a a a a a a a a a        A and 11 12 13 21 22 23 31 32 33 b b b b b b b b b        B . Write down in full the first column of the product AB and show that this can be put in the form 11 1 21 2 31 3b b b c c c , where 11 1 21 31 a a a        c , 12 2 22 32 a a a        c and 13 3 23 33 a a a        c . Write down corresponding expressions for the second and third column of AB. Hence show that the rank of AB cannot be greater than the rank of A. For the case where 1 2 2 1 2 5 5 3 3 3 5                       A , ,    , show that (i) for all values of  and  the rank of A is not greater than 2, (ii) if 0  and 1  , then, for all 3 3 matrix B, there are at least two linearly independent solutions for x of the equation ABx 0 , 3x  . (1993 A Level / FM / Jun / P1) www.KiasuExamPaper.com 709

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 4 of 17 11 Determine the rank of the matrix 1 1 2 3 3 2 3 13 4 4 9 7 11 9 17 36            A . Deduce that if x is a solution of the equation 1 1 2 3 2 3 4 4 9 11 9 17 p q r                                      Ax , where p, q and r are given real numbers, then 2 11 5 p q r               x , where   . Hence, or otherwise, for solution                x of the equation 4 8 17 37            Ax , (i) find x such that 0  , (ii) show that there is no x for which 2 2 2 2 1       . (1992 A Level / FM / Nov / P1) 12 The linear transformations 4 4 1T :   , 4 4 2T :   and 4 4 3T :   are represented by the matrices 1M , 2M and 2 1M M respectively, where 1 1 4 5 8 0 4 1 5 1 3 0 2 0 1 1 0              M and 2 1 2 1 3 1 0 4 5 3 2 7 13 1 4 6 1           M . (i) Show that the rank of 1M is equal to 3. (ii) Write down a basis for 1R , the range space of 1T , and find a basis for the null space of 1T . (iii) Find a basis for 2K , the null space of 2T , and hence show that 2K is a subspace of 1R . (iv) Hence, or otherwise, find three linearly independent vectors in the null space of 3T . (1997 A Level / FM / Nov / P1) www.KiasuExamPaper.com 710

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 5 of 17 13 Consider the equation Ax b where 1 2 1 3 2 1 5 7 4 3 7 1 3 14 15 43           A , 1 2 3 4 x x x x            x , 1 2 3 4 b b b b            b , and the elements of x and b are real. For the case where 1 2 3 4 0b b b b    , the set of solutions of x is denoted by K, and for the case where 1 3b  , 2 1b  , 3 7b  and 4 11b  , the set of solutions for x is denoted by S. (i) Show that K is a vector space and find its dimension. (ii) Show that S is not a vector space. (iii) Given that 1e and 2e are two linearly independent vectors belonging to K, show that S is the set of vectors of the form 1 2 1 1 1 1               e e , where  and  are real parameters. (1991 A Level / FM / Jun / P1) 14 Show that the set S if vectors given by 1 1 0 0 0 1 1 0, , ,1 0 0 0 0 0 1 1 S                                                    forms a basis for the linear space 4 . The linear transformation 3 4 L :   is defined by L 2 y zx x zy x yz x y z                     . Find the null space of L, and state its dimension. Show that 1 1 0 01 0 1 1 0L 1 2 1 2 1 0 0 00 0 0 1 1                                                         , and express 1 L 0 1         and 0 L 1 1         each as a linear combinations of the vectors of S. (1985 A Level / FM / Jun / P1) www.KiasuExamPaper.com 711

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 6 of 17 15 The linear transformation 4 4 T :   is represented by the matrix 2 2 1 2 3 4 2 5 4 5 3 5 2 7 3 9 6 12 2 14 3 18 a a a a a a                A