F07 - Matrices and Linear Spaces - Tutorial Set 3

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 1 of 18 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 2T :   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 674

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 3) Page 2 of 18 4 The linear transformation 3 3L:   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 Show that n B B for all positive integers n. 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 675

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and L