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...

A video lesson showing a series of proofs that culminate in explaining that any member of Rn can be represented as a unique sum of a member of some subspace of Rn and a member of its orthogonal complement. Ideas included in the proof are...

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 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]

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

A video lesson showing a series of proofs that culminate in explaining that any member of Rn can be represented as a unique sum of a member of some subspace of Rn and a member of its orthogonal complement. Ideas included in the proof are...

This video shows that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V.

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.

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