Linear Transformations

Explore the fundamental concepts of linear transformations in this comprehensive 7-hour course. Delve into topics such as graph representation, nullspace properties, and linear independence. Examine the Rank Nullity Theorem, Linear Extension Theorem, and coordinate systems. Investigate matrix operations, including multiplication and inverse matrices. Study isomorphisms, change of coordinates, and practical applications like reflections and projections. Learn how to integrate using coordinate transformations and discover the connections between linear algebra and calculus through derivative analysis.

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.