How do I exactly project a vector onto a subspace?

Question

I am trying to understand how - exactly - I go about projecting a vector onto a subspace. Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not sure if I am reading too much into this, or if this is something that I have missed. For a class I am taking, the proff is saying that we take a vector, and 'simply project it onto a subspace', (where that subspace is formed from a set of orthogonal basis vectors). Now, I know that a subspace is really, at the end of the day, just a set of vectors. (That satisfy properties [here](http://en.wikipedia.org/wiki/Projection_(linear_algebra))). I get that part - that its this set of vectors. So, how do I "project a vector on this subspace"? Am I projecting my one vector, (lets call it a[n]) onto ALL the vectors in this subspace? (What if there is an infinite number of them?) For further context, the proff was saying that lets say we found a set of basis vectors for a signal, (lets call them b[n] and c[n]) then we would project a[n] onto its [signal subspace](http://en.wikipedia.org/wiki/Signal_subspace). We project a[n] onto the signal-subspace formed by b[n] and c[n]. Well, how is this done exactly?..

Answer 1

Once you have a set of orthogonal basis representing a subspace, then you can project any vector with the same dimension as your subspace to you subspace. Projecting a vector into a subspace is the act adding up all the vectors that align with your basis subspace. Than means, you can write the projection of a vector v onto a subspace spanned by the orthogonal set of vectors {a_n} as:

{a_n}

proj_{{a_n}}(v) = (a₁·v/||a₁||²) a₁ + (a₂·v/||a₂||²) a₂+ ... + (a_n·v/||a_n||²) a_n

As an example, think about a plane spanned by two orthogonal 3 dimensional vectors in real euclidean space of 3 dimensions (ℜ³ ), then an arbitrary vector in ℜ³ can be projected onto this plane (subspace) by the process described above.

For further utilities, you can compose a matrix A, whose column vectors are the orthonormal basis of the subspace {a_n} (that means if the basis were not normalized, you have to normalize before constructing the matrix), then the projector to the subspace P = AA^T. To convince yourself of this, you can first check that P has the property of a projector (idemptotence), i.e. P²=P. You can work out the details of how the projection described in the equation above is captured in the matrix vector multiplication Pv.