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I've been following Gil Strang's lectures and he shows that $Null(A)$ is orthogonal to $Row(A)$ of an $m\times n$ matrix $A$ from the fact that matrix multiplication $Av$ is like taking the dot product of $v$ with the rows of $A$, or $Av=\begin{bmatrix}r_1v\\\vdots\\r_mv\end{bmatrix}$. If $v$ is in the null space, every $r_iv=0$, so $v$ must be orthogonal to the span of the rows of $A$. He uses a similar logic to show orthogonality of $Col(A)$ and $Null(A^T)$.

To me this only makes sense if $v$ and $r_i$ are seen as vectors in the standard basis. If $v$ has coordinates based on some arbitrary basis of $R^n$ (I don't even know what coordinates $r_i$ would be in terms of in this case), the dot product of the coordinates of two orthogonal vectors may not equal 0. However, $Av = 0$ ($Null(A)$) is still $v$ such that every $r_iv=0$. Is there a more general way to see orthogonality of the fundamental subspaces that is basis independent?

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  • $\begingroup$ The definition of orthogonal is dot product zero. Basis doesn't have anything to do with it. $\endgroup$ – Gerry Myerson Aug 14 at 7:37
  • $\begingroup$ @GerryMyerson From this question, I got the impression that two orthogonal vectors have an inner product of zero regardless of basis and computed by the dot product formula in the standard basis. If the basis is changed, the inner product is computed some other way (and usually not the same formula as the dot product) to get the same value (0 for two orthogonal vectors). Is this incorrect? $\endgroup$ – Yandle Aug 15 at 3:25
  • $\begingroup$ First, you define an inner product on the vector space; then, you define orthogonality to mean, inner product is zero. The definition of the inner product may or may not involve any particular basis. If it does involve some particular basis (for example, if it is given by the dot product on ${\bf R}^n$), and you choose to represent elements of the vector space with respect to some other basis, then, yes, you have to change the way of computing the inner product. $\endgroup$ – Gerry Myerson Aug 15 at 4:53
  • $\begingroup$ @GerryMyerson My question is linked to when the way the inner product is computed is basis dependent. The $v$ such that $Av=0$ is when the dot product of the $v$ with each row of $A$ is zero regardless what basis is used to represent the coordinates of $v$, but $v$ may be represented with respect to some basis where the dot product of two orthogonal vectors is not zero (but inner product is). This is why I am confused about how null and row spaces are orthogonal if $v$ is represented wrt some arbitrary basis. $\endgroup$ – Yandle 2 days ago
  • $\begingroup$ The null space and row space of a matrix are properties of the matrix and are completely independent of any choice of basis or of inner product. The dot product of a row in the matrix and an element of the null space is zero. Beyond that, it's all up for grabs. $\endgroup$ – Gerry Myerson yesterday

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