Khan Academy: A Projection Onto a Subspace Is a Linear Transforma

Showing that a projection onto a subspace is a linear 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...

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...

This video shows an example of finding the transformation matrix for the projection onto a subspace with an orthonormal basis.

A video lesson proving that the projection of a vector is actually the closest vector in the subspace to the original vector. [9:05]

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...

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...

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.

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]

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...