**Rotation Matrix**

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.

**Projection**

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 .