• Matrix Vector Products
  • Introduction to the Null Space of a Matrix
  • Null Space 2: Calculating the null space of a matrix
  • Null Space 3: Relation to Linear Independence
  • Column Space of a Matrix
  • Null Space and Column Space Basis
  • Visualizing a Column Space as a Plane in R3
  • Proof: Any subspace basis has same number of elements
  • Dimension of the Null Space or Nullity
  • Dimension of the Column Space or Rank
  • Showing relation between basis cols and pivot cols
  • Showing that the candidate basis does span C(A)
  • A more formal understanding of functions
  • Vector Transformations
  • Linear Transformations
  • Matrix Vector Products as Linear Transformations
  • Linear Transformations as Matrix Vector Products
  • Image of a subset under a transformation
  • im(T): Image of a Transformation
  • Preimage of a set
  • Preimage and Kernel Example
  • Sums and Scalar Multiples of Linear Transformations
  • More on Matrix Addition and Scalar Multiplication
  • Linear Transformation Examples: Scaling and Reflections
  • Linear Transformation Examples: Rotations in R2
  • Rotation in R3 around the X-axis
  • Unit Vectors
  • Introduction to Projections
  • Expressing a Projection on to a line as a Matrix Vector prod
  • Compositions of Linear Transformations 1
  • Compositions of Linear Transformations 2
  • Linear Algebra: Matrix Product Examples
  • Matrix Product Associativity
  • Distributive Property of Matrix Products
  • Linear Algebra: Introduction to the inverse of a function
  • Proof: Invertibility implies a unique solution to f(x)=y
  • Surjective (onto) and Injective (one-to-one) functions
  • Relating invertibility to being onto and one-to-one
  • Determining whether a transformation is onto
  • Linear Algebra: Exploring the solution set of Ax=b
  • Linear Algebra: Matrix condition for one-to-one trans
  • Linear Algebra: Simplifying conditions for invertibility
  • Linear Algebra: Showing that Inverses are Linear
  • Linear Algebra: Deriving a method for determining inverses
  • Linear Algebra: Example of Finding Matrix Inverse
  • Linear Algebra: Formula for 2x2 inverse
  • Linear Algebra: 3x3 Determinant
  • Linear Algebra: nxn Determinant
  • Linear Algebra: Determinants along other rows/cols
  • Linear Algebra: Rule of Sarrus of Determinants
  • Linear Algebra: Determinant when row multiplied by scalar
  • Linear Algebra: (correction) scalar muliplication of row
  • Linear Algebra: Determinant when row is added
  • Linear Algebra: Duplicate Row Determinant
  • Linear Algebra: Determinant after row operations
  • Linear Algebra: Upper Triangular Determinant
  • Linear Algebra: Simpler 4x4 determinant
  • Linear Algebra: Determinant and area of a parallelogram
  • Linear Algebra: Determinant as Scaling Factor
  • Linear Algebra: Transpose of a Matrix
  • Linear Algebra: Determinant of Transpose
  • Linear Algebra: Transposes of sums and inverses
  • Linear Algebra: Transpose of a Vector
  • Linear Algebra: Rowspace and Left Nullspace
  • Lin Alg: Visualizations of Left Nullspace and Rowspace
  • Linear Algebra: Orthogonal Complements
  • Linear Algebra: Rank(A) = Rank(transpose of A)
  • Linear Algebra: dim(V) + dim(orthogonoal complelent of V)=n
  • Lin Alg: Representing vectors in Rn using subspace members
  • Lin Alg: Orthogonal Complement of the Orthogonal Complement
  • Lin Alg: Orthogonal Complement of the Nullspace
  • Lin Alg: Unique rowspace solution to Ax=b