Projection is the process of decomposing a vector into two distinct parts one parallel with the subspace and on that is orthogonal to the subspace. The result always gives a unique decomposition.

Given a vector and a subspace we can decompose as follows

Where is parallel to and is orthogonal to it.

Properties

Given the projection of a vector into and the following are true:

• is the closest vector to in .
• The distance from to is .

Calculating

Given a subspace generated from the columns of a matrix and a vector for any solution to the equation

then .