{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern