Ace Linear Algebra

Notation of matrices

Solving systems of linear equations by graphing

Representing linear system as a matrix

The three types of matrix row operations

Solving a linear system with matrices using Gaussian elimination

Row reduction and echelon forms

Linear combination and vector equations in $R^n$

Matrix equation Ax=b

Solution sets of linear systems

Application of linear systems

Linear independence

Image and range of linear transformations

Properties of linear transformation

The matrix of a linear transformation

Adding and subtracting matrices

Multiplying a matrix by a scalar

Multiplying a matrix by another matrix

Zero matrix

Identity matrix

Properties of matrix addition

Properties of matrix scalar multiplication

Properties of matrix to matrix multiplication

The determinant of a 2 x 2 matrix

The determinant of a 3 x 3 matrix (General & Shortcut Method)

Properties of determinants

Solving linear systems using Cramer's Rule

2 x 2 invertible matrix

The inverse of a 2 x 2 matrix

Solving linear systems using 2 x 2 inverse matrices

The inverse of a 3 x 3 matrices with matrix row operations

The invertible matrix theorem

Properties of subspace

Column space

Null space

Dimension and rank

Eigenvalues and eigenvectors

The characteristic equation

Diagonalization

Complex eigenvalues

Discrete dynamical systems

Inner product, length, and orthogonality

Orthogonal sets

Orthogonal projections

Least-squares problem

Applications to linear models