Question about projections and subspaces

[JamesGoh]

If we want to caculate the projection of a single vector, v=(1,2) (which is an element of an R2 vector space called V) onto the subspace of V (which we call W), do we use

projection of v onto W = <v,w1>w1 + <v,w2>w2 + ... <v,wn>wn

However, if the individual values of v are not known (that is v=(x,y) ), do we calculate the matrix of projection ?

that is, we do A(A$^{T}$A)$^{-1}$A$^{T}$

If we have to determine the matrix of projection, is it because we don't know what x and y is, so the safe bet is to determine the matrix of projection ?

 

[HallsofIvy]

I really have no idea exactly what are asking. If you have a subspace, W, of an inner product space, V, and $\{w_1, w2, ..., v_n\}$ is an orthonormal basis for W, then, yes, the projection of v, a vector in v, onto W is $<v, w_1>w_1+ <v, w_2>w_2+ ...+ <v, w_n>wn$. If you v is a "general" vector, then, yes, there exist a matrix such that multiplying any vector v by that matrix giives the projection. That is obviously true because projection is a linear operator. But I have no idea what you mean by "$A(A^TA)^{-1}A^T$" because I do not know what "A" is. Where did you get the matrix A?

 

[JamesGoh]

sorry A is the matrix that is formed from the basis vectors in subspace W

where W = { basis 1, basis 2 }