EE-Learning
Showing that A-transpose x A is invertible
Catogry:
Math
Subject:
Linear Algebra
Course:
Matrix Transformations
Lecture List
Showing that A-transpose x A is invertible
rank(a) = rank(transpose of a)
Visualizations of left nullspace and rowspace
Rowspace and left nullspace
Transpose of a vector
Transposes of sums and inverses
Transpose of a matrix product
Determinant of transpose
Transpose of a matrix
Determinant as scaling factor
Determinant and area of a parallelogram
Simpler 4x4 determinant
Upper triangular determinant
Determinant after row operations
Duplicate row determinant
Determinant when row is added
(correction) scalar multiplication of row
Determinant when row multiplied by scalar
Rule of Sarrus of determinants
Determinants along other rows/cols
n x n determinant
3 x 3 determinant
Formula for 2x2 inverse
Example of finding matrix inverse
Deriving a method for determining inverses
Showing that inverses are linear
Simplifying conditions for invertibility
Matrix condition for one-to-one transformation
Exploring the solution set of Ax = b
Determining whether a transformation is onto
Relating invertibility to being onto and one-to-one
Surjective (onto) and injective (one-to-one) functions
Proof: Invertibility implies a unique solution to f(x)=y
Introduction to the inverse of a function
Distributive property of matrix products
Matrix product associativity
Matrix product examples
Compositions of linear transformations 2
Compositions of linear transformations 1
Expressing a projection on to a line as a matrix vector prod
Introduction to projections
Unit vectors
Rotation in R3 around the x-axis
Linear transformation examples: Rotations in R2
Linear transformation examples: Scaling and reflections
More on matrix addition and scalar multiplication
Sums and scalar multiples of linear transformations
Preimage and kernel example
Preimage of a set
im(T): Image of a transformation
Image of a subset under a transformation
Linear transformations as matrix vector products
Matrix vector products as linear transformations
Linear transformations
Vector transformations
A more formal understanding of functions