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Orthogonal complements
Catogry:
Math
Subject:
Linear Algebra
Course:
Alternate Coordinate Systems (bases)
Lecture List
Showing that an eigenbasis makes for good coordinate systems
Eigenvectors and eigenspaces for a 3x3 matrix
Eigenvalues of a 3x3 matrix
Finding eigenvectors and eigenspaces example
Example solving for the eigenvalues of a 2x2 matrix
Proof of formula for determining eigenvalues
Introduction to eigenvalues and eigenvectors
Gram-Schmidt example with 3 basis vectors
Gram-Schmidt process example
The Gram-Schmidt process
Orthogonal matrices preserve angles and lengths
Example using orthogonal change-of-basis matrix to find transformation matrix
Finding projection onto subspace with orthonormal basis example
Projections onto subspaces with orthonormal bases
Coordinates with respect to orthonormal bases
Introduction to orthonormal bases
Changing coordinate systems to help find a transformation matrix
Alternate basis transformation matrix example part 2
Alternate basis transformation matrix example
Transformation matrix with respect to a basis
Invertible change of basis matrix
Change of basis matrix
Coordinates with respect to a basis
Another least squares example
Least squares examples
Least squares approximation
Projection is closest vector in subspace
Another example of a projection matrix
Subspace projection matrix example
A projection onto a subspace is a linear transformation
Visualizing a projection onto a plane
Projections onto subspaces
Rowspace solution to Ax = b example
Unique rowspace solution to Ax = b
Orthogonal complement of the nullspace
Orthogonal complement of the orthogonal complement
Representing vectors in rn using subspace members
dim(v) + dim(orthogonal complement of v) = n
Orthogonal complements