Vector 12 Week 3 Learning Goals

Here are the knowledge and skills you should master by the end the third week.

12.1 Linear Transformations and Matrix Inverses

I should be able to do the following tasks:

Determine whether a mapping from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) is a linear transformation.
Use the RREF of the corresponding matrix to determine whether \(T(\mathsf{x})\) is one-to-one and/or onto.
Describe 2D linear transformations as a mixture of geometric operations, including expansion, contraction, reflection, rotation, shearing and dimension reduction.
Perform a 2D translation using 3D homogeneous coordinates.
Multiply an \(m \times n\) matrix with an \(n \times p\) matrix to get an \(m \times p\) matrix.
Determine whether a \(2 \times 2\) matrix is invertible.
Find the inverse of a \(2 \times 2\) matrix by hand.
Use RStudio to check for invertiblity and to find the inverse of an \(n \times n\) square matrix.
Explain the connection between Gaussian Elimination, elementary matrices, and the matrix inverse.

I should know and be able to use and explain the following terms or properties.

linear transformation: \(T(a \mathsf{u} + b \mathsf{v}) = a T(\mathsf{u}) + b T(\mathsf{v})\)
domain, codomain (aka target) and range (aka image)
\(T\) maps vector \(\mathsf{x}\) to its image \(T(\mathsf{x})\)
one-to-one
onto
standard matrix for a linear transformation
homogeneous coordinates
transpose of a matix
invertible matrix
elementary matrices

I should understand and be able to explain the following concepts:

A linear transformation \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m\) corresponds to multiplication by an \(m \times n\) matrix \(A\).
\(T(\mathsf{x})=\mathsf{A} \mathsf{x}\) is a one-to-one linear transformations if and only \(\mathsf{A}\) has linearly independent columns
\(T(\mathsf{x})=\mathsf{A} \mathsf{x}\) is an onto linear transformations if and only if the columns of \(\mathsf{A}\) span \(\mathbb{R}^m\).
The Invertible Matrix Theorem (Section 2.3, Theorem 8, page 112) is one of the highlights of the course! It gives 12 different conditions that all equivalent! You should think deeply about why everything comes together like this for square matrices.