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Linear Transformation

Vectors should be stored in columns.

A m\times n (m rows and n columns) matrix \mathbf{A} can be seen as a linear transformation of a n dimensional space into m dimensional space. It is a mapping from n-dimensional space to m-dimensional space.

The n column vectors in \mathbf{A} are the coordinates of the vectors in the new m-dimensional space. During the transformation of the space, the the basis of the n dimensional space are mapped to these vectors in the new space.

So a column vector \mathbf{x} which original was a point in the n-dimensional space, would get its new coordinate in the m-dimensional space as \mathbf{Ax}. In this multiplication, sum of element-wise multiplication are calculated between \mathbf{x} and each row of \mathbf{A}.

Each of the n columns use the m unit vectors (basis in the new space) to represent its coordinate. The ith row of \mathbf{A}, which has n elements, represents how long these n basis vectors are on the ith dimension of the m dimensions. So this multiplication between \mathbf{x} and the ith row would calculate the length of \mathbf{x} in the ith dimension.

Matrix multiplication can be understand as a sequence of linear transformations of spaces. For example, \mathbf{ABC} can be seen as: first transform by \mathbf{C}, then by \mathbf{B}, finally by \mathbf{A}. They read from right to left like f(g(x)), which first calculate g.