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Hi:

I see an example about nullspace and orthogonality, the example is following:

$$Ax=\begin{bmatrix} 1 & 3 &4\\ 5 & 2& 7 \end{bmatrix} \times \left[ \begin{array}{c} 1 \\ 1\\-1 \end{array} \right]=\begin{bmatrix} 0\\0\end{bmatrix}$$

The conclusion says the nullspace of [itex]A^T[/itex] is only the zero vector(orthogonal to every vector). I don't know why the columns of A and nullspace of [itex]A^T[/itex] are orthogonal spaces.

I know nullspace is the solution of Ax=0; but in this theorem, why columns of A is related

to nullsapce of [itex]A^T[/itex].

Thanks.

I see an example about nullspace and orthogonality, the example is following:

$$Ax=\begin{bmatrix} 1 & 3 &4\\ 5 & 2& 7 \end{bmatrix} \times \left[ \begin{array}{c} 1 \\ 1\\-1 \end{array} \right]=\begin{bmatrix} 0\\0\end{bmatrix}$$

The conclusion says the nullspace of [itex]A^T[/itex] is only the zero vector(orthogonal to every vector). I don't know why the columns of A and nullspace of [itex]A^T[/itex] are orthogonal spaces.

I know nullspace is the solution of Ax=0; but in this theorem, why columns of A is related

to nullsapce of [itex]A^T[/itex].

Thanks.

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