Asked • 05/20/19

How do I exactly project a vector onto a subspace?

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?..

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