Systems of Linear Equations. Row Operations

Echelon Forms. Gaussian Elimination

Matrices and matrix Operations

Inverses. Transpose

Elementary Row Operations

Systems of Equations and Invertibility

Determinants: Definition and Computation

Determinants Using Row Reduction

Determinants: Properties

Vector, Norms, Dot Product

Lines and Planes

Cross Product

Vector Spaces. Some Proofs.

Linear Independence. Dimension and Basis.

Some Important Theorems for Bases in Vector Spaces and Dimension.

Change of Basis.

Row, Column, and Null Spaces.

Rank and Nullity.

Eigenvalues and Eigenvectors.

Diagonalizable Matrices.

Best Approximation. Least Squares

Orthogonal Matrices.

Diagonalization of Symmetric Matrices.

Singular Value Decomposition.

Linear Transformations.