F07 - Matrices and Linear Spaces - Tutorial Set 2

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 2) Page 1 of 8 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Matrices and Linear Spaces (Tutorial Set 2) This tutorial set is for the following sections from the notes: §5 Real Vector Spaces §6 Span, Linear Independence, Basis and Dimension §7 Row Space, Column Space and Null Space Basic Mastery Questions 1 Determine whether each of the following is a vector space. Justify your answer. (i)  : 0 has no solutionS ax b ax b    (ii)   3 :T    u a u 0 where a is a fixed nonzero vector in 3 , (iii) d: 0d yU y x y x       . (iv) Let V be the set of all functions f :   . We define addition and scalar multiplication on V as follows: For f ,g V and k  ,       f g f g x x x   ,     f fk x k x  . (v) 3 with addition  and scalar multiplication  defined as  u v 0 for any 3, u v  ; k u 0 for any 3 u  , k  . (vi)      1 0 all continuous functions f : 0,1 such that f d 0 W x x   . 2 Show that each of the following subsets of 3 is a vector space. Find also a basis for each of these subspaces. (i)   , , : , , 2 ,U x y z x t y t z t t      . (ii)   , , : 2 3 0W x y z x y z    . 3 Show that each of the following subsets of 2P is a vector space. Find also a basis for each of these subspaces. (i)   22 : ,S a bx ax a b     . (ii)     2f : f 0 has two real roots 2 and 1T x x   P . www.KiasuExamPaper.com 651

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 2) Page 2 of 8 4 Determine by observation whether each of the following subsets of 3 is a subspace of 3 . (i)   , 3 , 2 :a a a a   , (ii)    2 2 2, , : 0x y z x y z    , (iii)   , , : 2 5 1a b c a b c   , (iv)   , , : 2x y z x y  , (v)   ,1,1 :a a   , (vi)   , , :x y z x y z  , (vii)   , , : 0a b c a b c   , (viii)   , , :a b c a b c  . 5 Prove that the set of vectors       1,0, 1 , 0, 2,0 , 1,1,1 is both a linearly independent and a spanning set of 3 . Determine, with reasons, whether each of the following sets of vectors forms a basis for V or not: (i)       1,0, 1 , 0, 2,0 , 1,1,1 , (ii)       1,0, 1 , 0, 2,0 , 0,0,0 , (iii)         1,0, 1 , 0, 2,0 , 1,1,1 , 0,1,0 , (iv)     0, 2,0 , 1,1,1 . (1977 A Level / FM / Nov / P2) 6 (a) Let V and W be subspaces of the linear space n . Show that the set of vectors of the form v w , where Vv and Ww , is a subspace of n . (b) For each of the following subsets of 3 determine whether or not it is a subspace, giving reasons for your answers. Find a basis for each subset which you consider to be a subspace. (i)    3 1 2 3 1, , : 1x x x x   , (ii)    3 1 2 3 1 2 3, , :x x x x x x   , (iii)    3 1 2 3 1 2, , : 7x x x x x   , (iv)    3 2 2 1 2 3 1 2, , : 0x x x x x    . (1978 A Level / FM / Jun / P2) www.KiasuExamPaper.com 652

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 2) Page 3 of 8 7 Let U and W be subspaces of a vector space V. Give a counterexample to show that U W need not be a subspace of V. 8 (a) Enlarge the following sets of linearly independent vectors to form bases. (i) Enlarge     1,2,1 , 1, 2,1 to form a basis for 3 . (ii) Enlarge 1 3 1 3 1 0, ,1 0 0 0 1 0                     to form a basis for  2,2M  . (b) Reduce the following spanning sets to form bases. (i) Reduce   2 2 21 , 2 ,2 ,1x x x x x x     to form a basis for 2P . (ii) Reduce 2 0 1 1 1 , 1 , 0 , 1 0 2 1 1                                            to form a basis for : 2 0 x y x y z z                . Practice Questions 9 Let A be a fixed 2 2 matrix, and let   2,2 :W   X M AX XA . Prove that W is a subspace of  2,2M  . 10 (a) Explain a subspace of 2 can only be  0 , a line through the origin, or 2 . (b) Explain a subspace of 3 can only be  0 , a line through the origin, a plane through the origin, or 3 . 11 Given that 1 , 1 , 1 b a a b b a                                 is not a basis for 3 , prove that 3 3 3 1 0a ab b    . (1984 A Level / FM / Jun / P2) 12 Find a basis for the vector space spanned by the vectors  1, 2, 1 ,  3, 1,2 ,  2, 10,8 ,  7, 7,8 . What is the dimension of this space? For what value (or values) of a does  1, 4,a belong to the space? (1976 A Level / FM / Jun / P2) www.KiasuExamPaper.com 653

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 2) Page 4 of 8 13 The vectors 1 2 3, ,b b b form a basis for a vector space V, and 1 1 2 2 3 3    a b b b is a vector in V, 1 2 3, ,   being scalars. Obtain a necessary and sufficient condition that the vectors 2 3, ,a b b are linearly independent. A vector x in V is such that 1 1 2 2 3 3    x b b b , where scalars 1 2 3, ,   are positive. If 2 3, ,a b b are linearly independent and x is a linear combination of these vectors with nonnegative scalar coefficients, show that 1 0  . Show, by a counterexample, that if 1 0  , then the coefficients expressing x as a linear combination of 2 3, ,a b b need not be nonnegative. (1975 A Level / FM / Nov / P2) 14 Let V and W be subspaces of n . Show that V W is also a subspace of n . Find a basis for V W in the following cases: (i)    4 1 2 3 4 1 3, , , : 0V x x x x x x    ,    4 1 2 3 4 4 2, , , : 2W x x x x x x   . (ii)    4 1 2 3 4 1 2, , , : 2 0V x x x x x x    ,    4 1 2 3 4 4 2, , , : 7 3W x x x x x x   . (iii) 3 1 2 1: , where 0 1 3V           x Ax 0 A , 3 2 1 7: , where 4 2 14W             x Bx 0 B . (1979 A Level / FM / Jun / P1) 15 The set 2P consists of all polynomials in x, of degrees less than or equal to 2, and having real coefficients, i.e.   2 2 : , ,ax bx c a b c   P  . Show that , with usual operations of addition and multiplication by a real number, 2P is a linear (vector) space over  , of dimension 3. For each of the following subsets of 2P , determine whether or not it is a subspace, giving brief reasons for your answers. Give a basis for each subset which you consider to be a subspace. (a)     2f : f 0 0x  P , (b)     2f : f 0 1x  P , (c)     2f : f 1 0x  P , (d)       2f : f f for all x x x x   P  . (1980 A Level / FM / Jun / P1) www.KiasuExamPaper.com 654

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 2) Page 5 of 8 16 In each of the following cases, state whether, with usual operations, the given set forms a linear space over the field  . For each of those which you consider to be a linear space, give a basis for the space, and for each of those which you consider not to be a linear space, justify your answer. (a) 2 1 2 d dsolutions of 3 2 0d d y yS yx x         . (b) 2 2 2 d dsolutions of 3 2 2 3d d y yS y xx x          . (c)   3 , : 2 1S x y y x   . (d)   4 , , : 0S x y z x y z    . (d)  5 : i, ,S z z a b a b     (i.e.  5S z   ). (1982 A Level / FM / Nov / P1) 17 S is a subspace of 4 spanned by vectors         1,0,1,1 , 5,0, 2, 2 , 2,0,1,1 , 3,0,0,0 , and T is a subspace of 4 spanned by vectors         1, 1,0,1 , 0,0,0,0 , 0,1,1,0 , 0,3,3,