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Math Help - True/False about Vector Spaces

  1. #1
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    True/False about Vector Spaces

    Hello, i have a true false question about vector spaces that i cannot figure out and its part of our test review so i need to know it.

    A.)The columns of an invertible nxn matrix form a basis for R^n

    B.)In some cases, the linear dependence relations amoung the columns of a matrix can be affected by certain elementary row operations on the matrix.

    C.)A single vector by itself is linearly dependent

    D.)if H=Span(b1,.....,bp) then (b1,.....,bp) is a basis for H

    E.)A basis is a spanning set that is as large as possible.

    I think that A,B,C are true. Am i right? and are the others true?


    Thank You,
    Diggidy
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Diggidy View Post
    I think that A,B,C are true. Am i right?
    More important, show some work. Why do you think they are true?. For example: A invertible implies \det A\neq 0 wich implies \textrm{rank}A=n etc.


    and are the others true?

    A little help: for D) choose H=\mathbb{R}^2 and b_1=(1,0),b_2=(2,0),b_3=(0,1) .
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  3. #3
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    Well (C) is false, unless the vector is the 0 vector (why?).

    For (B), I want to say that that is also false (i guess it depends what "affect" means).

    D and E are false as well, both with simple explanations. What happens if you repeat a vector in a spanning set? What is the largest spanning set you could possibly take?
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