Suppose is a 1-dimensional subspace of (so that ). Then given , we define the projection of onto to be Note that this quantity makes sense, as implies .
The vector depends only on the vector and the subspace . In other words, it does
not depend on the choice of basis for .
- Proof
- Any other basis vector for can be written as for some . Then
The vector represents the -component of (in texts, this projection is also referred to as the component of in the direction of . We prefer the subspace interpretation, as it makes clear the independence on the choice of basis element).