Linear Algebra

Lower Division College

1 Introduction to Vectors
2 Vector Addition
3 Scalar Multiplication

4 Drawing Vector

5 Dot Products
6 Vector Lengths
7 Vector Spaces
8 Linear Independence of Vectors

9 Vector Space Dimensions

10 Matrices, Addition and Scalar Multiplication

11 Matrix Multiplication

12 Fundamental Question: \(A\overrightarrow{v} = \overrightarrow{b}\)
13 Idea of Elimination
14 Identity and Inverse Matrices
15 Finding Inverses Using Row Reduction

16 Elimination and Factorization

17 Transpositions and Permutations

18 Vector Spaces and Subspaces

19 The Nullspace of a Matrix

20 The Rank and Row Reduced Form

21 Solutions to \(Ax = b\)

22 Independence, Basis, and Dimension

23 Dimension of the Four Subspaces

24 Orthogonality

25 Projections

26 Least Square Approximations

27 Orthogonal Bases and Gram-Schmidt

28 Determinants

29 Permutations and Cofactors

30 Cramer's Rule

31 Eigenvalues and Eigenvectors

32 Diagonalizing a Matrix

33 Symmetric Matrices

34 Positive Definite and Positive Semi-Definite