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). 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. We project a[n] onto the signal-subspace formed by b[n] and c[n]. Well, how is this done exactly?..
Thanks in advance, let me know if I can clarify anything!
P.S. I appreciate your help, and I would really like for the clarification to this problem to be somewhat 'concrete' - for example, something that I can show for myself over MATLAB. Analogues using 2-D or 3-D space so that I can visualize what is going on would be very much appreciated as well.