## Vector Spaces and Subspaces

- Overview of Vector Spaces and Axioms
- Common Vector Spaces and the Geometry of Vector Spaces
- Example using three of the Axioms to prove a set is a Vector Space
- Overview of Subspaces and the Span of a Subspace- Big Idea!
- Common Examples of Subspaces and Recap of how to Prove Subspaces
- Examples #1 & #2 – are a line and parabola a Subspace?
- Examples #3 & #4 – show the set is or is not a Subspace
- Example #5 is w in the subspace spanned by the given vectors?
- Example #6 show the set is or is not a Subspace
- Example #7 show the set is or is not a Subspace using the Span
- Examples #8 & #9 – show the set is or is not a Subspace
- Example #10 is w in the subspace spanned by the given vectors?
- Examples #11 & #12 – show the set is or is not a Vector Space
- Example #10 for what value of h is y in the subspace spanned by the given vectors?

## Null, Column, and Row Spaces

- Definition of the Null Space and Observations about the Null Space
- Row Space and Column Space and Theorems and Definitions about the Nul, Col and Row
- Example #1 – Find the Basis and Dimensions for the Null Space, Column Space and Row Space
- Example #2 – Find the Basis and Dimensions for the Null Space, Column Space and Row Space
- Examples #3 & 4 – Find a matrix A such that W = ColA
- Examples #5 & 6 – Show the given set is a vector space
- Example #7 – is W in the NulA or the ColA?
- Example #8 – is W in the NulA or the ColA?
- Example – find the explicit description of the Null Space by finding the vectors that Span the Null Space
- Example – find k such that NulA and ColA are subspaces of k-dimensional space
- Kernel and Range of a Linear Transformation
- Example – what form do the vectors in the kernel look like?
- Recap of Essential Elements of the Null Space and Column Space
- Example – Find the Basis and Dimensions for the Null Space, Column Space and Row Space

## Linearly Independent Sets and Bases

- Overview of the Linearly Independent and Dependent Sets
- Examples – determine if the set is Linearly Independent or Dependent
- Theorem and Definition for a Basis and Spanning Set
- Example #1 – is this a Linearly Independent or Dependent Set
- Example #2 – is this a Linearly Independent or Dependent Set
- Example #3 – is the set a Standard Basis?
- Example #4 – is the set a Basis?
- Example #5 – Explain why the set is not a Basis?
- The Spanning Set Theorem with Examples
- Example – find bases for the Null Space and Column Space
- Example – find bases for the Null Space and Column Space
- Example – find a basis for the space spanned by the set of vectors
- Example – is the set of polynomial functions a basis?

## Coordinate Systems and The Dimensions of a Vector Space

- Unique Representation Theorem and the B-Coordinates of X Matrix
- Example #1 B-Coordinate Matrix
- Change of Coordinate Matrix from B to the Standard Basis
- Example #2 Change-of-Coordinate Matrix
- Example #3 Change-of-Coordinate Matrix
- Example #4 Change-of-Coordinate Matrix
- Overview of Isomorphism
- Standard Basis for Polynomials
- Example #1 – Coordinate Vectors to Determine if Linearly Independent
- Example #2 – Do the set of polynomials form a basis?
- Dimensions of a Vector Space
- Example #1 – Expand the Spanning Set to form a Basis
- Example #2 – Find a Basis and if necessary expand the Spanning Set
- Example #3 – Do the polynomials form a Basis?
- Example #4 – Find the dimNul and dimCol
- Example #5 – find a basis for the set of polynomials
- Example #6 – find change-of-coordinate matrix for the set of polynomials

## Rank

- Rank and Nullity Theorem and Definitions
- Example #1 – Find the NulA, ColA, RowA, and dimensions
- Examples #2 & 3 – What is the Rank or Null Space of the given matrix by inspection only
- Examples #4 & 5 – What is the Rank or Null Space of the given matrix by inspection only
- Examples #6 & 7 – What is the Rank or Null Space of the given matrix by inspection only
- Adding onto the Invertible Matrix Theorem
- True/False – Three Examples for an m x n matrix and the NulA, ColA, RowA

## Change of Basis

- Change of Basis Theorems and Definitions
- Example #1 & 2 – Find the Change-of-Coordinate Matrix and C-Coordinate Vector
- Change-of-Coordinate Matrix from B to C Theorems and Definitions
- Example #3 & 4 – Find the Change-of-Coordinate Matrix from B to C
- Example #5 – find the Change-of-Coordinate Matrix for Polynomials

## Markov Chain Applications

- Overview of Markov Chains and Chance Process
- Transition Matrices vs Migration Matrices
- Example #1 – set up a Transition Matrix for College Acceptance
- Example #2 – set up a Transition Matrix for Pizza Delivery
- Example #3 – interpret the Transition Matrix
- Example #4 – Rent-a-Lemon and the Migration Matrix
- Finding the Steady-State Vector with Example
- Example – Find the Steady-State Vector for Population Change