Consider as shown below where I have my data in vector $y$ and some functions in $A$ and their coefficients $c$ in a vector $x$.
The problem was to find the function that estimates some data. I have solved this by virtue of $Ax=b$ so to find $b$ I can compute the pseudo inverse and the coefficients are found by $A^+b=x$ where $A^+$ is the pseudo inverse. Thus I have a matrix $A$ to estimate $y$ with coefficients in $x$.
The above method works. My question:
How can I do this using a projection matrix? This is my attempt: To make $b$ unto the $range(A)$ it can be projected down using a projection matrix $P$. Thus it can be stated $Pb=Ax$
Then it can be stated $A^+Pb=x$ However this did not work as the dimensions did not check out. So how would this correctly be done? The formula for the projection matrix I used was $P=A(A^*A)^{-1}A^*$. Also the book I am following stated there are many ways to find the least-squares solution either by QR, Cholesky, SVD, $Pb=Ax$ and the one I used $A^+b=x$.