Module 5: Linear Transformations
Study linear maps between vector spaces, their matrix representations, and the fundamental relationship between kernel, range, and dimension.
Module Lessons
1
Linear Transformations: Definition and Examples
Define linear transformations, verify linearity, and explore key examples including rotations, reflections, and projections.
2
The Matrix of a Linear Transformation
Every linear transformation from R^n to R^m corresponds to multiplication by a unique m x n matrix.
3
Kernel and Range
Define the kernel (null space) and range (image) of a linear transformation and compute them.
4
The Rank-Nullity Theorem
Prove and apply the fundamental theorem connecting the dimensions of kernel and range.