Monday, September 18, 2006

Linear Algebra For Graphics Geeks (The Column View)

Finding myself in the mood for a brief return to linear algebra.

This is post #8 of Linear Algebra for Graphics Geeks. Here are links to posts #1, #2, #3, #4, #5, #6, #7.

The point of this post is something called "the column view" of matrices and matrix products. Very frequently, linear algebra offers multiple perspectives of the same mathematical phenomenon; each perspective offers new insight; "the column view" is such a perspective.

Consider the transformation of the vector [x y z] by the following matrix.

The "column view" offers the insight that the product is a linear combination of the columns of the matrix, each column is weighted by x, y and z, respectively.

If the N columns of a square matrix form a basis for RN, then the dimension of the column space is N. At the other extreme, if every element of the matrix is zero, the only possible result of any product is a zero-dimensional result, all zeroes.

In the 3D case, if the matrix is all zeroes, the product is [0 0 0] regardless of the value of [x y z]. Given a matrix with N columns, the possible dimensionality of transformed vectors can be anywhere in the range from 0 to N.

This is one of those aspects of linear algebra that can be a duh! or a revelation depending on one's experiences.

From the perspective of the "column view," the identity matrix is an orthonormal basis for Rn. In the case of R3, below, the first column is a unit vector along the x-axis, the second column is a unit vector along the y-axis and finally the last column is a unit vector along the z-axis. The effect of the identity matrix on vectors, of course, is no change.

From the "column view" perspective, it's quite clear you can go anywhere in R3 via combinations of the columns of the identity matrix--every transformation is a combination of the columns of the identity matrix and any point in R3 going into the transformation is the same coming out of the transformation.

If we completely zero out one of the columns in the identity matrix by replacing a one with a zero, the effect is quite clear: we'll lose a dimension. For example, in the following, you can see that every [x y z] will be projected onto the x-y plane regardless of the value of z. In this case, the column space is 2D.

Likewise, if we zero yet another column, we'll reduce the dimensionality of the column space down to 1D, with the effect being every vector being reduced to its x component.

Finally, we've already discussed the case where a matrix is all zeroes.

A key point here is that when we transform a vector with a matrix, the resulting vector is always in the column space of the matrix; the dimensionality of the column space determines the dimensionality of the results.

Next: The Pseudoinverse.


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