• vectors and their properties
• linear combinations of vectors
• solutions of linear systems as linear combinations of vectors
• the linear span of a collection of vectors
• when is a given vector an element of the linear span of a vectors
• the relationship between linear span and echelon form
• matrix vector multiplication
• matrix equations Ax=b
• linear independence of vectors
• linear span and linear independence
• homogeneous systems
• algebra of matrix vector multiplication
• particular plus homogeneous solutions
• The Unifying Theorem - Version 1. (Theorem 2.21)

• linear transformtions, domain, codomain, range (or image), null space (or kernel)
• "one-to-one" and "onto" linear transformations
• Unifying Theorem, Version 2
• matrix addition and scalar multiplication
• matrix multiplication
• the identity matrix
• matrix algebra
• matrix transpose
• composition of linear transformations and matrix multiplication
• powers of a matrix
• diagonal, upper triangular, and lower triangular matrices
• elementary matrices and elementary row operations
• partitioned matrices
• the inverse of a linear transformation and inverse matrices
• computing matrix inverses
• Unifying Theorem, Version 3

Key Concepts:

• Chapter 2:
  • linear combinations and linear span
  • the relationship between linear span and echelon form
  • the relationship between linear independence and echelon form
  • particular plus homogeneous solutions
  • The Unifying Theorem - Version 1
• Chapter 3
  • the relationship between linear transformations and matrices
  • properties of "one-to-one" and "onto" linear transformations
  • all the versions of the Unifying Theorem
  • matrix algebra
  • composition of linear transformations and matrix multiplication
  • matrix inverses and their computation
  • elementary row operations as matrix products

Skills to Master:

• Chapter 2
  • Solving matrix equations
  • writing the solution set of a linear system as a linear combination of vectors
  • Checking linear independence and the relationship to echelon form
  • computing particular and homogeneous solutions
• Chapter 3
  • testing if a matrix is one-to-one or onto
  • computing an elementary matrix
  • computing a matrix inverse
  • applying the Unifying Theorem