Linear Transformations

This course develops linear transformations through nullspaces, ranges, rank-nullity, bases, coordinates, matrix representations, inverses, projections, and changes of coordinates.

Syllabus

Linear Transformations.
Graph of a linear transformation.
Nullspace is a subspace.
Are derivatives linearly independent ?.
Range and Linear Independence.
One to one and linear independence.
Symmetrization.
Rank Nullity Theorem.
Linear Extension Theorem.
Linear transformation with given range.
Coordinates.
Derivative in a box.
Matrix of a matrix.
Matrix with respect to a basis.
Pseudo diagonalization.
Matrix Multiplication.
Matrix Multiplication Derivation.
ABBA.
A^2 = A.
A^2 = I.
A^2 = 0.
Isomorphism.
Left and right inverse.
Dimension and Isomorphism.
Inverse Matrix.
Inverse Transformation.
Miracle of Linear Algebra.
Linear Algebra Derivative.
Change of coordinates.
Change of Matrix.
Reflection about y = 2x.
Projection on y = 2x.
Integrate via change of coordinates.