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Vectors, Span and Linear Combinations
Introduction to multidimensional vectors
Multidimensional Vectors Examples
Parametric Representations Of Lines
Linear Combinations and Span
Matrix Vector Products
Null Space, Column Space and Basis
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)
Subspaces
Linear Subspaces
Basis of a Subspace
Linear Independence
Introduction to Linear Independence
More on linear independence
Span and Linear Independence Example
Orthogonality
Orthogonal Complements
PROOF : dim(V) + dim(orthogonoal complelent of V)=n
Representing vectors in Rn using subspace members
Orthogonal Complement of the Orthogonal Complement
Orthogonal Complement of the Nullspace
Unique rowspace solution to Ax=b
Rowspace Solution to Ax=b example
Projections
Projections onto Subspaces
Visualizing a projection onto a plane
PROOF : A Projection onto a Subspace is a Linear Transformation
Subspace Projection Matrix Example
Another Example of a Projection Matrix
Change of Basis
Coordinates with Respect to a Basis
Change of Basis Matrix
Invertible Change of Basis Matrix
Transformations
Transformation Matrix with Respect to a Basis
Alternate Basis Tranformation Matrix Example
Alternate Basis Tranformation Matrix Example Part 2
Changing coordinate systems to help find a transformation matrix
Orthonormal Basis
Introduction to Orthonormal Bases
Coordinates with respect to orthonormal bases
Projections onto subspaces with orthonormal bases
Finding projection onto subspace with orthonormal basis example
Example using orthogonal change-of-basis matrix to find transformation matrix
Orthogonal matrices preserve angles and lengths
Gram-Schmidt Process
The Gram-Schmidt Process
Gram-Schmidt Process Example
Gram-Schmidt example with 3 basis vectors
Eigenvalues and Eigenvectors
Introduction to Eigenvalues and Eigenvectors
Proof of formula for determining Eigenvalues
Example solving for the eigenvalues of a 2x2 matrix
Finding Eigenvectors and Eigenspaces example
Eigenvalues of a 3x3 matrix
Eigenvectors and Eigenspaces for a 3x3 matrix
Showing that an eigenbasis makes for good coordinate systems

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