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I am confused over the geometric meaning of columns and rows in a matrix. My understanding from 3Blue1Brown's Linear Algebra series was that the elements of a matrix column are the coordinates which the tip of a single basis vector lands on at the end of the linear transformation, while the elements of a matrix row are the coefficients of an equation (if the matrix is augmented) or expression (if the matrix is not augmented). However, I am now studying matrix subspaces, and it appears that both columns and rows can be treated as vector tip coordinates to solve for different subspaces.

Is there a second vector geometric interpretation of a matrix where the elements of a row are the coordinates on which the tip of a single basis vector lands (which would presumably be identical to applying the original geometric interpretation to the matrix's transpose)? If so, what is the geometric or conceptual difference between matrix columns and rows, which justifies the differences in how we operate on them, and how we relate them to linear systems of equations?

user10478
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1 Answers1

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Actually, 3Blue1Brown gives you that interpretation as well, though he doesn't go deep enough to get to this particular aspect. In Chapter 7, he discusses duality - how linear transformations into a 1D line correspond to specific vectors in space, And how when expressed as matrices, the transformation matrix was the column vector flipped over on its side - i.e., converted into a row. These "row vectors" are the "dual vectors" of the normal column vectors. And you can consider linear transformations in terms of row vectors in a very similar fashion to how the videos talk about them in terms of column vectors.

A basis provides a way to identify each vector in space with a specific dual vector. (In the videos he makes it seem like there is one natural to make this assignment - but that is because he has a natural basis, $\hat i, \hat j, \hat k$ that he uses. And covering this more generally was outside the scope of what he was trying to do.) When you transpose a matrix, you are actually making use of this vector-dual vector identification to change your transformation to act on the dual vectors instead of the original vectors.

So, the point is, you can consider either the columns or the rows to be vectors. In either case, the matrix converts vectors of the same form into other vectors of the same form. But when acting on columns, the matrix multiplies in on the left of the column, while when acting on rows, it multiplies in on the right of the row: $$\begin{bmatrix}a & b\\c & d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}w\\z\end{bmatrix}$$ $$\begin{bmatrix}p&q\end{bmatrix}\begin{bmatrix}a & b\\c & d\end{bmatrix} = \begin{bmatrix}r&s\end{bmatrix}$$

Paul Sinclair
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