1. Scalars, Vectors, Matrices

2. Systems of Linear Equations

3. Matrix Notation and Operations

4. Row Echelon Form & Reduced Row Echelon Form (RREF)

5. Gaussian Elimination & Gauss-Jordan Elimination

Determinants 

1. Determinant Properties

2. Cofactor Expansion

3. Minors and Cofactors

4. Cramer's Rule

5. Applications of Determinants

Linear Transformations 

1. Definition and Examples

2. Matrix Representation of Linear Transformations

3. Kernel and Image

4. Isomorphisms

5. Change of Basis

6. Similarity of Matrices

Inner Product Spaces 

1. Dot Product

2. Norms and Distance

3. Orthogonality

4. Orthogonal Projection

5. Gram-Schmidt Process

6. Orthonormal Bases

7. QR Decomposition

Advanced Topics 

1. Matrix Addition, Subtraction, Multiplication

2. Transpose of a Matrix

3. Identity and Inverse Matrices

4. Diagonal, Symmetric, and Skew-Symmetric Matrices

5. Elementary Matrices

6. Block Matrices

Vector Spaces 

1. Definition and Axioms

2. Subspaces

3. Linear Independence

4. Span and Basis

5. Dimension

6. Null Space, Column Space, Row Space

Eigenvalues and Eigenvectors 

1. Characteristic Polynomial

2. Diagonalization

3. Eigenspaces

4. Spectral Theorem (for symmetric matrices)

5. Defective and Non-defective Matrices

6. Jordan Canonical Form (Advanced)

Matrix Factorizations 

1. LU Decomposition

2. QR Decomposition

3. Cholesky Decomposition

4. Singular Value Decomposition (SVD)

Applications