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Math Help - Kernel is orthogonal complement

  1. #1
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    Kernel is orthogonal complement

    Let L:Rn -> Rm be linear transformation with matrix A. Show that ker(L) is orthogonal complement of row space of A.
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  2. #2
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    Re: Kernel is orthogonal complement

    Suppose x\in Ker(A). Then 0=(Ax,y)=(x,A^{T}y) means that Ker(A)^{\bot}=Ran(A^{T}). If we represent A by its rows w_{i}, hence A_{T} by its columns: A^{T}=(w_{1}|w_{2}|...|w_{m}), each row being an n-dimensional vector, and each y as y=\sum_{i=1}^{i=n}{c_{i}e_{i}}, then A^{T}y=\sum_{i=1}^{i=n}{c_{i}A \cdot e_{i}=\sum_{i=1}^{i=n}{c_{i}w_{i}}}, which means that Ran(A^{T})=LinSpace({w_{i}}).
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