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Math Help - Dimension of and Basis for Solution Set of Homogeneous System

  1. #1
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    Dimension of and Basis for Solution Set of Homogeneous System

    This is probably pretty simple I just want to make sure I'm doing it right:

    Find the dimension of and a basis for the solution set of the following homogeneous systems of equations:

    1. x1+2x2-3x3+x4=0


    2.x1+2x2=0
    x1-x2=0


    Any help would be appreciated! Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Dimension of and Basis for Solution Set of Homogeneous System

    Quote Originally Posted by divinelogos View Post
    Find the dimension of and a basis for the solution set of the following homogeneous systems of equations: 1. x1+2x2-3x3+x4=0
    Solving the system:

    \begin{bmatrix}{x_1}\\{x_2}\\{x_3}\\{x_4} \end{bmatrix}=\begin{bmatrix}{-2\lambda+2\mu-\gamma}\\{\lambda}\\{\mu}\\{\gamma}\end{bmatrix}= \lambda \begin{bmatrix}{-2}\\{1}\\{0}\\{0}\end{bmatrix}+\mu \begin{bmatrix}{2}\\{0}\\{1}\\{0}\end{bmatrix}+ \gamma \begin{bmatrix}{-1}\\{0}\\{0}\\{1}\end{bmatrix}

    Those three vectors span the subspace and are linearly independent. Try 2.
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