Vector Spaces and Subspaces
1 hr 24 min 15 Examples
- Overview of Vector Spaces, Axioms, and Geometry (Example #1)
- Proving a Set is a Vector Space (Example #2)
- Subspaces, Span, and Big Ideas (Example #3)
- Proving Subspaces and Common Examples (Example #4)
- Line and Parabola as Subspaces? (Examples #5 & #6)
- Is it a Subspace? (Examples #7 & #8)
- Is w in the Subspace Spanned? (Example #9)
- Subspaces and Span (Examples #10 – #13)
Null, Column, and Row Spaces
1 hr 19 min 12 Examples
- Null Space Definition and Observations (Example #1)
- Row and Column Spaces (Example #2)
- Basis and Dimensions (Examples #3 & #4)
- Matrix A with W = ColA (Examples #5 & #6)
- Is W in NulA or ColA? (Examples #7 & #8)
- Null Space Vectors and k-Dimensional Spaces (Examples #9 & #10)
- Kernel, Range, and Linear Transformations (Examples #11 & #12)
Linearly Independent Sets and Bases
1 hr 9 min 10 Examples
- Linearly Independent and Dependent Sets Overview (Example #1)
- Basis, Spanning Set, and Theorems (Example #2)
- Linear Independence and Dependence (Examples #3 & #4)
- Standard Basis and Bases (Examples #5 – #7)
- Spanning Set Theorem and Examples (Example #8)
- Bases for Null Space and Column Space (Examples #9 & #10)
Coordinate Systems and The Dimensions of a Vector Space
1 hr 23 min 12 Examples
- Unique Representation Theorem and B-Coordinates (Example #1)
- Change of Coordinate Matrix and Standard Basis (Examples #2 – #4)
- Isomorphism Overview and Standard Basis for Polynomials (Examples #5 & #6)
- Dimensions, Vector Space, and Basis (Examples #7 – #10)
- Change-of-Coordinate Matrix for Polynomials (Example #11)
Rank
30 min 7 Examples
- Rank and Nullity Theorem (Example #1)
- Rank and Null Space by Inspection (Examples #2 – #5)
- Invertible Matrix Theorem and True/False Examples (Examples #6 & #7)
Change of Basis
28 min 5 Examples
- Change of Basis Theorems (Examples #1 & #2)
- Change-of-Coordinate Matrix from B to C (Examples #3 & #4)
- Change-of-Coordinate Matrix for Polynomials (Example #5)
Markov Chain Applications
46 min 5 Examples
- Markov Chains and Transition vs. Migration Matrices (Examples #1 & #2)
- Interpreting Transition Matrix and Rent-a-Lemon (Examples #3 & #4)
- Finding the Steady-State Vector with Population Change Example
Chapter Test
1 hr 8 min 11 Problems
- Subspace, Span, Linear Independence, and Basis (Problems #1 & #2)
- ColA, NulA, and Basis for RowA, ColA, NulA (Problems #3 – #5)
- B-Coordinates, Change Coordinate Matrix, and Linear Independence (Problems #6 – #9)
- Steady-State Vector and True or False (Problems #10 & #11)
