The rotation matrix
rotates points in the -plane counterclockwise through an angle about the origin.
For example rotating the vector 45 degrees counterclockwise gives us:
Finding Least Squares Solution
Given (inconsistent system), solve
instead to get a least squares solution of the original equation.
If we know a least squares solution of , we can find the projection of onto the column space of by
Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)
If is a matrix with columns, then
(=number of pivot columns,
=number of non-pivot columns.)
Linear Independence and the Wronskian
A set of vector functions from to is linearly independent in the interval if for at least one value of in the interval .