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Math Help - Linear Algebra Basis and Span Questions.

  1. #1
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    Linear Algebra Basis and Span Questions.

    Hi all, im revising for finals and the following questions are causing me problems.

    Q1) Determine a basis for each of the following subspace of R^3

    The parts that confuse me are

    the line: x = 2t, y = t, z = 4t
    the plane x - y = 0
    all vectors of the form (x; y; z), where y = x + z

    I understand the basis but I cant seem to understand these??

    Q2) W
    e have stated that [1; x; x^2; ......; x^n] is a basis for Wn, where Wn denotes the vector space of polynomials of maximal degree n. Verify that [1; x -1; x^2 - x; : : : ; x^n -x^n-1] is also a basis for Wn

    Now for this i was generally thinking of subbing in values for the relevant x's but is this correct?

    Q3)Let V be a vector space with basis [v1; v2; v3]. Which of the following
    collections are also a basis for V ?

    1) [v1, v1 + v2, v1 + v3]
    2) [2v1,3v2, v1 + v2];
    3) [v1, v2 + v3, v1 -v3, v1 + v2]
    4) [v2, v3 - v, v3 + v2]
    5) [ v2, v3- v1]
    6) [v1 - v2 + 2v3 , 2v2 + v3, 3v1 + v2 -3v3]

    Again is it easier to just make values for v1,v2,v3 up ?
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  2. #2
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    Re: Linear Algebra Basis and Span Questions.

    Quote Originally Posted by Chewybakas View Post
    Hi all, im revising for finals and the following questions are causing me problems.

    Q1) Determine a basis for each of the following subspace of R^3

    The parts that confuse me are
    Are you really clear on what a basis is?

    the line: x = 2t, y = t, z = 4t
    Any vector is of the form <x, y, z>= <2t, t, 4t>= t<2, 1, 4>. Can you answer the question now?

    the plane x - y = 0
    The equation is the same as y= x
    Any vector is of the form <x, y, z>= <x x, z>= <x, x, 0>+ <0, 0, z>= x<1, 1, 0>+ z<0, 0, 1>

    all vectors of the form (x; y; z), where y = x + z

    Try this one yourself now.

    I understand the basis but I cant seem to understand these??

    Q2) W
    e have stated that [1; x; x^2; ......; x^n] is a basis for Wn, where Wn denotes the vector space of polynomials of maximal degree n. Verify that [1; x -1; x^2 - x; : : : ; x^n -x^n-1] is also a basis for Wn

    Now for this i was generally thinking of subbing in values for the relevant x's but is this correct?

    Q3)Let V be a vector space with basis [v1; v2; v3]. Which of the following
    collections are also a basis for V ?

    1) [v1, v1 + v2, v1 + v3]
    2) [2v1,3v2, v1 + v2];
    3) [v1, v2 + v3, v1 -v3, v1 + v2]
    4) [v2, v3 - v, v3 + v2]
    5) [ v2, v3- v1]
    6) [v1 - v2 + 2v3 , 2v2 + v3, 3v1 + v2 -3v3]

    Again is it easier to just make values for v1,v2,v3 up ?
    I guess it would but then it would be even easier to replace the entire problem with "1+ 1"! You are given v1, v2, and v3. You are told that v1, v2, v3 is a basis. Any basis for an n dimensional vector space has three properties: the vectors span space, the vectors are independent, and there are n vectors. And any two of those implies the third.

    Here, all of the given collections have three vectors so it is sufficient to show that each v1, v2, v3 can be written as a linear combination of the given vectors.
    Last edited by HallsofIvy; April 24th 2013 at 02:40 PM.
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