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Math Help - subspace and basis

  1. #1
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    subspace and basis

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  2. #2
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    Re: subspace and basis

    sure we'll help...where are you stuck...show us what you've done so far.
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  3. #3
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    Re: subspace and basis

    The first problem is to show this is a subspace. Do you know what "subspace" means? What properties must a set have to be a subspace?

    The second and third ask you to find a basis. Do you know what a "basis" is?
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  4. #4
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    Re: subspace and basis

    1)showing that W is a subspace was easy (closed under addition and scalar multiplication)
    2)I took [1 2 0] and [0 0 1] as a basis( dimension 2) because [ a 2a b] is a linear combination of these two but is it correct or should I've done something else?
    3)for the third question I setup the system
    1 0 1 0|0
    1 0 0 1|0
    0 1 0 0|0
    and deduced the linearly independent vectors which form the basis for R3
    so was my work correct?
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  5. #5
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    Re: subspace and basis

    for 2) you showed spanning, but not linear independence.

    you only need three equations.

    you want to find a 3rd vector (x,y,z) such that {(1,2,0),(0,0,1),(x,y,z)} is linearly independent.

    one way to do this is to pick (x,y,z) so that it is orthogonal to (1,2,0) and (0,0,1).

    if (1,2,0).(x,y,z) = 0, then x+2y = 0. that is: y = -x/2.

    if (0,0,1).(x,y,z) = 0, then z = 0. can you combine these two conditions? prove the resulting set of 3 vectors is linearly independent.

    the system you set up makes no sense at all, your vectors live in R3, why do you have a matrix with 4 columns?
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