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Math Help - Problem finding a basis for a given subspace

  1. #1
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    Problem finding a basis for a given subspace

    Ok, so I need to find a basis for subspace A.

    A = {(x1, x2, x3) E R^3 | 2x1 + x3 = 0, x1 + x2 - x3 = 0} of R^3

    I understand how to determine whether something is a basis of something else, but I cannot wrap my head around how to determine my own basis for this subspace.
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  2. #2
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    Quote Originally Posted by crymorenoobs View Post
    Ok, so I need to find a basis for subspace A.

    A = {(x1, x2, x3) E R^3 | 2x1 + x3 = 0, x1 + x2 - x3 = 0} of R^3

    I understand how to determine whether something is a basis of something else, but I cannot wrap my head around how to determine my own basis for this subspace.
    Set x_1=t then you have a system of two equations in two unknowns.

    Using the first equation you get

    x_3=-2t

    Using the 2nd equation we get
    x_2=-x_1+x_3=-t-2t=-3t

    Using this we get

    \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} =\begin{bmatrix} t \\ -2t \\ -3t \end{bmatrix}=t\begin{bmatrix} 1\\ -2 \\ -3 \end{bmatrix}

    So any multiple of the last vector will space that space.
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