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
.