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101st_tut3

# 101st_tut3 - denotes the maximum of a and b(d Let v 2 V and...

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MATH1111/2010-11/ Tutorial III 1 MATH1111 Tutorial III 1. Consider the vectors a = (1 1 1) T ; b = (2 0 1) T ; c = ( 1 1 2) T ; d = (1 2 1) T : (a) Reduce the matrix a b c d in row echelon form. (b) Could one decide by part (a) whether a ; b ; c ; d form a spanning set for R 3 ? How? What else can you tell from the matrix? (c) Find a basis from a ; b ; c ; d for R 3 . (d) Do a ; b form a basis for R 3 ? If no, could you add suitable vector(s) other than c or d to get a basis? 2. Let V be a vector space and dim V = 7. Is it true that for every integer r = 0 ; 1 ; 2 ; ; 7, one can always nd a subspace W r of V such that dim W r = r ? 3. Let S; T be subspaces of a vector space V . Prove or disprove the following. (a) S = T if and only if dim S = dim T . (b) dim( S \ T ) min(dim S; dim T ) where min( a; b ) denotes the minimum of a and b . (c) dim( S [ T ) max(dim S; dim T ) where max( a; b ) denotes the maximum of
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Unformatted text preview: ) denotes the maximum of a and b . (d) Let v 2 V and U = Span( v ). Then dim( U + S ) = dim S + 1. 4. Given three ordered bases for R 3 : E = [ e 1 ; e 2 ; e 3 ], F = [ f 1 ; f 2 ; f 3 ] and H = [ h 1 ; h 2 ; h 3 ], where e 1 ; e 2 ; e 3 form the standard basis, and f 1 = (1 1 1) T ; f 2 = (1 2 2) T ; f 3 = (2 3 4) T ; h 1 = (3 2 5) T ; h 2 = (1 1 2) T ; h 3 = (2 3 2) T : (a) If [ x ] F = (1 ± 1 2) T , ±nd [ x ] E and [ x ] H . (b) Find, by evaluating [ f 1 ] H , [ f 2 ] H , [ f 3 ] H , the transition matrix from F to H . (c) Find the transition matrix from F to H by ±rst evaluating the transition matrices from F to E and from H to E ....
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