Orthogonal complements

  1. Orthogonal complements

  2. dim(v) + dim(orthogonal complement of v) = n

  3. Representing vectors in rn using subspace members

  4. Orthogonal complement of the orthogonal complement

  5. Orthogonal complement of the nullspace

  6. Unique rowspace solution to Ax = b

  7. Rowspace solution to Ax = b example


Orthogonal projections

  1. Projections onto subspaces

  2. Visualizing a projection onto a plane

  3. A projection onto a subspace is a linear transformation

  4. Subspace projection matrix example

  5. Another example of a projection matrix

  6. Projection is closest vector in subspace

  7. Least squares approximation

  8. Least squares examples

  9. Another least squares example


Change of basis

  1. Coordinates with respect to a basis

  2. Change of basis matrix

  3. Invertible change of basis matrix

  4. Transformation matrix with respect to a basis

  5. Alternate basis transformation matrix example

  6. Alternate basis transformation matrix example part 2

  7. Changing coordinate systems to help find a transformation matrix


Orthonormal bases and the Gram-Schmidt process

  1. Introduction to orthonormal bases

  2. Coordinates with respect to orthonormal bases

  3. Projections onto subspaces with orthonormal bases

  4. Finding projection onto subspace with orthonormal basis example

  5. Example using orthogonal change-of-basis matrix to find transformation matrix

  6. Orthogonal matrices preserve angles and lengths

  7. The Gram-Schmidt process

  8. Gram-Schmidt process example

  9. Gram-Schmidt example with 3 basis vectors



  1. Introduction to eigenvalues and eigenvectors

  2. Proof of formula for determining eigenvalues

  3. Example solving for the eigenvalues of a 2×2 matrix

  4. Finding eigenvectors and eigenspaces example

  5. Eigenvalues of a 3×3 matrix

  6. Eigenvectors and eigenspaces for a 3×3 matrix

  7. Showing that an eigenbasis makes for good coordinate systems

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