Khan Academy: Orthogonal Complements: Orthogonal Complement of the Nullspace

Video introducing the orthogonal complement of a subspace and showing that it is also a subspace. Proves that the null space of a matrix is equivalent to the orthogonal complement of its row space by showing that every member of the null...

A video proving that the orthogonal complement of the orthogonal complement of some subspace V must be a member of the original subspace V.

Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n

This video shows that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=b.

An instructional video that gives an example of how to find a solution to the equation Ax = b. In the example, the matrix A is 2 x 2. Shows that when the solution is a member of the row space of A for a given vector b, a member of the...

Video builds upon the column spaces and null spaces of the 2x3 matrix and its transpose found in the previous video. Uses them to explain some relationships between the four fundamental subspaces, namely that the orthogonal complement of...

Sal shows an example finding the relative complement or difference of two sets A and B.

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.

Finding the difference between sets and complement of a set.

Video introducing the orthogonal complement of a subspace and showing that it is also a subspace. Proves that the null space of a matrix is equivalent to the orthogonal complement of its row space by showing that every member of the null...