Matrix transformations videos

A more formal understanding of functions

Vector transformations

Linear transformations as matrix vector products

Matrix vector products as linear transformations

Linear transformations

im(T): Image of a transformation

Image of a subset under a transformation

Preimage and kernel example

Preimage of a set

Sums and scalar multiples of linear transformations

More on matrix addition and scalar multiplication

Linear transformation examples: Scaling and reflections

Linear transformation examples: Rotations in R2

Introduction to projections

Rotation in R3 around the x-axis

Unit vectors

Expressing a projection on to a line as a matrix vector prod

Matrix product examples

Compositions of linear transformations 2

Compositions of linear transformations 1

Matrix product associativity

Distributive property of matrix products

Proof: Invertibility implies a unique solution to f(x)=y

Introduction to the inverse of a function

Surjective (onto) and injective (one-to-one) functions

Determining whether a transformation is onto

Relating invertibility to being onto and one-to-one

Exploring the solution set of Ax = b

Simplifying conditions for invertibility

Matrix condition for one-to-one transformation

Showing that inverses are linear

Deriving a method for determining inverses

Example of finding matrix inverse

3 x 3 determinant

n x n determinant

Determinants along other rows/cols

Rule of Sarrus of determinants

Determinant when row multiplied by scalar

Determinant when row is added

(correction) scalar multiplication of row

Duplicate row determinant

Upper triangular determinant

Determinant after row operations

Determinant and area of a parallelogram

Simpler 4x4 determinant

Determinant as scaling factor

Transpose of a matrix

Determinant of transpose

Transposes of sums and inverses

Transpose of a matrix product

Transpose of a vector

Rowspace and left nullspace

Visualizations of left nullspace and rowspace

rank(a) = rank(transpose of a)

Showing that A-transpose x A is invertible