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Linear Algebra Topics

1Linear Equations with Matrices

1.1Notation of matrices
1.2Solving systems of linear equations by graphing
1.3Representing linear system as a matrix
1.4The three types of matrix row operations
1.5Solving a linear system with matrices using Gaussian elimination
1.6Row reduction and echelon forms
1.7Linear combination and vector equations in RnR^n
1.8Matrix equation Ax=b
1.9Solution sets of linear systems
1.10Application of linear systems

2Linear Transformation

2.1Linear independence
2.2Image and range of linear transformations
2.3Properties of linear transformation
2.4The matrix of a linear transformation

3Matrix Operations

3.1Adding and subtracting matrices
3.2Multiplying a matrix by a scalar
3.3Multiplying a matrix by another matrix
3.4Zero matrix
3.5Identity matrix
3.6Properties of matrix addition
3.7Properties of matrix scalar multiplication
3.8Properties of matrix to matrix multiplication

4Inverse of Matrices

4.1The determinant of a 2 x 2 matrix
4.22 x 2 invertible matrix
4.3The inverse of a 2 x 2 matrix
4.4Solving linear systems using 2 x 2 inverse matrices
4.5The inverse of a 3 x 3 matrices with matrix row operations
4.6The invertible matrix theorem

5Subspace of Rn\Bbb{R}^n

5.1Properties of subspace
5.2Column space
5.3Null space
5.4Dimension and rank

6Determinant of a Matrix

6.1The determinant of a 3 x 3 matrix (General & Shortcut Method)
6.2Properties of determinants
6.3Solving linear systems using Cramer's Rule

7Eigenvalue and Eigenvectors

7.1Eigenvalues and eigenvectors
7.2The characteristic equation
7.3Diagonalization
7.4Complex eigenvalues
7.5Discrete dynamical systems

8Orthogonality and Least Squares

8.1Inner product, length, and orthogonality
8.2Orthogonal sets
8.3Orthogonal projections
8.4Least-squares problem
8.5Applications to linear models
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