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Math Help - Kernel

  1. #1
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    Kernel

    I found [T]E to be:


    2 2 4 8
    0 0 -4 -12
    0 0 2 6
    0 0 0 0

    but how can I find the kernel kerT? Appreciate any help!
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  2. #2
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    The kernel of a linear transformation is the solution of Ax=0, or simply the set of vectors that T maps into 0. Can you find it now?
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  3. #3
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    That much I knew... I'm just not seeing how to apply it to this problem. Is it the set of vectors that maps T[E] into 0? Thank you for replying.
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  4. #4
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    Then use that definition!

    T_E(v)=  \begin{bmatrix} 2 & 2 & 4 & 8 \\ 0 & 0 & -4 & -12 \\ 0 & 0 &  2 & 6 \\ 0 & 0 & 0 & 0\end{bmatrix}\begin{bmatrix}u \\ x\\ y \\ z\end{bmatrix} = \begin{bmatrix}2u+ 2x+ 4y+ 8z \\-4 y+ 12 z\\ 2y+ 2z \\ 0 \end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \\ 0\end{bmatrix}

    which gives the three equations 2u+ 2x+ 4y+ 8z= 0, -4y- 12z= 0, and 2y+ 6z= 0. What values of u, x, y, and z satisfy those equations?
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