F07 - Matrices and Linear Spaces - Tutorial Set 2 (modified)

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 666

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 3 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 e ach subset which you consider to be a subspace. (i)    3 1 2 3 1, , : 1x x x x 