Khan Academy
Khan Academy: Linear Algebra: Projections Onto Subspaces
A video lesson defining projections onto subspaces. Uses the definition for a projection onto a line and then shows that a generalized version of the formula works for a projection onto any subspace. Includes a concrete example and...
Khan Academy
Khan Academy: Linear Algebra: Projection Is Closest Vector in Subspace
A video lesson proving that the projection of a vector is actually the closest vector in the subspace to the original vector. [9:05]
Khan Academy
Khan Academy: Linear Algebra: Projection Onto a Subspace Is a Lin Transformation
A video lesson proving that any projection onto a subspace is actually a linear transformation. Also, if given the basis for the subspace then the specific transformation matrix for the projection can be found. Includes a brief...
Khan Academy
Khan Academy: Linear Algebra: Least Squares Approximation
A video lesson explaining the least squares approximation for otherwise unsolvable matrix equations. Presents the motivation for why the least squares approximation is useful. Derives the formula for finding the least squares...
Khan Academy
Khan Academy: Linear Algebra: Subpsace Projection Matrix Example
An instructional video that works through a concrete example for how to find the projection of an arbitrary vector onto a specific subspace in R4. Uses a 4 x 2 basis matrix for the subspace. Matrix operations are used along with the...
Khan Academy
Khan Academy: Linear Algebra: The Gram Schmidt Process
Video about the Gram-Schmidt process which is used to create an orthonormal basis from any basis. First an orthonormal basis is created in one dimension by normalizing the vector and then in two dimensions by building on the...
Khan Academy
Khan Academy: Linear Algebra: Projection Onto Subspace With Orthonormal Basis Ex
Video defining a specific orthonormal basis and using the simplified formula found in the previous video to calculate the projection of a vector onto a subspace with the given orthonormal basis. [6:42]
Khan Academy
Khan Academy: Linear Algebra: Visualizing a Projection Onto a Plane
A video lesson showing what a projection onto a plane could look like. Illustrates that the newly derived definition of a projection holds true for projections onto subspaces other than lines. [9:27]
Khan Academy
Khan Academy: Linear Algebra: Projections Onto Subspaces With Orthonormal Bases
Video giving another reason why orthonormal bases are useful. First reviews how to find the projection of a vector onto a subspace and then describes how an orthonormal basis can be used to find the projection. Simplifies the...
Khan Academy
Khan Academy: Linear Algebra: Another Example of a Projection Matrix
A video lesson working another example for finding the projection of an arbitrary vector onto a specific subspace in R3. The basis of the subspace is a 3 x 2 matrix. Uses an alternate method for find the transformation matrix that...
Khan Academy
Khan Academy: A Projection Onto a Subspace Is a Linear Transforma
A video lesson proving that any projection onto a subspace is actually a linear transformation. It includes a brief description of how the results can be useful in 3-D graphical programming.
Khan Academy
Khan Academy: Orthogonal Projections: Least Squares Approximation
A video lesson explaining the least squares approximation for otherwise unsolvable matrix equations. Presents the motivation for why the least squares approximation is useful. Derives the formula for finding the least squares approximation.
Khan Academy
Khan Academy: Finding Projection Onto Subspace With Orthonormal Basis Example
This video shows an example of finding the transformation matrix for the projection onto a subspace with an orthonormal basis.
Khan Academy
Khan Academy: Projections Onto Subspaces With Orthonormal Bases
A video about projections onto subspaces with orthonormal bases.
Khan Academy
Khan Academy: Orthogonal Projections: Another Example of a Projection Matrix
A video lesson figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first.