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Math Help - find the basis

  1. #1
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    find the basis

    Let L be the line spanned by v_1=[1, 1, 1]. Find a basis {v_2, v_3} for the plane perpendicular to L, and verify that B= {v_1,, v_2, v_3} is a basis for R3.

    Let ProjL denote the projection onto the line L. Find the matrix B for ProjL with respect to the basis B.

    Can you just help me get started with the problem. I'll do the rest on my own.
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  2. #2
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    Quote Originally Posted by Taurus3 View Post
    Let L be the line spanned by v_1=[1, 1, 1]. Find a basis {v_2, v_3} for the plane perpendicular to L, and verify that B= {v_1,, v_2, v_3} is a basis for R3.

    Let ProjL denote the projection onto the line L. Find the matrix B for ProjL with respect to the basis B.

    Can you just help me get started with the problem. I'll do the rest on my own.
    Since v_1 is perpendicular to the plane you want to span it is the planes normal vector.

    Since the equation of the plane must pass though the origin it will have the form
    x+y+z=0

    Can you finish from here?
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  3. #3
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    ummm......actually no. Sorry. I mean I know how to find the eigenvalues and then the basis. But this question is weird.
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  4. #4
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    Quote Originally Posted by Taurus3 View Post
    ummm......actually no. Sorry. I mean I know how to find the eigenvalues and then the basis. But this question is weird.
    We don't need eigenvectors or eigenvalues.

    We need to find two vectors that span the above plane. e.g we need to find the basis for the null space of this matrix.

    \begin{bmatrix} 1 & 1 & 1& 0\end{bmatrix}

    If it helps you can think of it as this matrix with added rows of zeros.

    \begin{bmatrix} 1 & 1 & 1& 0 \\ 0 & 0 & 0& 0 \\ 0 & 0 & 0& 0 \\ \end{bmatrix}

    Since this is already in reduced row form We know that we have two free parameters. Let z=t,y=s then

    x+t+s=0 \iff x =-t-s

    So the basis of the nullspace is

    \begin{pmatrix} x \\ y \\ z\end{pmatrix} = \begin{pmatrix} -t-s \\ s \\ t\end{pmatrix} = \begin{pmatrix} -t \\ 0 \\ t\end{pmatrix}+ \begin{pmatrix} -s \\ s \\ 0\end{pmatrix} = \begin{pmatrix} -1 \\ 0 \\ 1\end{pmatrix}t+\begin{pmatrix} 0\\ -1 \\ 1\end{pmatrix}s

    You can verify that the two above vectors are perpendicular to the first.

    Can you finish from here?
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  5. #5
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    So for my 2nd question, would it just be B= [(1, 1, 1), (0, -1, 1), (-1, 0, 1)]?
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    So for my 2nd question, would it just be B= [(1, 1, 1), (0, -1, 1), (-1, 0, 1)]?
    No that is the matrix of the linear transformation from the standard basis to your new basis B.

    Hint: In your new basis T(v_1)=v_1 and T(v_2)=T(v_2)=0 So what is the matrix of this transformation?
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  7. #7
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    this is what I don't get. How do you know T(v1)=v1 and that T(v2)=0?
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  8. #8
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    Quote Originally Posted by Taurus3 View Post
    this is what I don't get. How do you know T(v1)=v1 and that T(v2)=0?
    Since \displatstyle T(\vec{x})=\text{proj}_{v_1}\vec{x}=\frac{\vec{v_1  }\cdot \vec{x}}{||v_1||^2}\vec{v_1}

    This is the definition of projecting one vector onto another. This will verify the above "claim".
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