Solution to $Ax =b$ Assume $A$ is a $m \times n$ matrix. We want to see whether the linear system $Ax=b$ has any solution for $x$ given $b$. One way to check this is:
"This linear system of equation has a solution if the b is contained in the column space of A."
1- Does anybody know a good reference for this?
2- How can we check that b is within the column space of A using projection? Any reference for this method?
 A: If you have your heart set on a reference Linear Algebra and Its Applications https://www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X
is a good textbook that deals extensively with the question of whether systems like that have solutions.
However, to more directly answer question 1, consider what the column space of matrix $A$ is. Consider A to consist of column vectors $[a_1,a_2...a_n]$, then the column space is the set of all vectors that can be expressed as a linear combination of these vectors.
Put in another way the column space of $A$ is the set of all vectors $b$ such that there exists are group of coefficients $x_1,x_2...x_n$ where $x_1a_1+x_2a_2+...+a_nx_n=b$.
This expression is actually equivalent to how we define the solution set of the equation $Ax=b$, so these two concepts are really the same by definition.
Projection does not seem to be what you are looking for. Conceptually you project a vector into a vector space (often a vector space that does not already contain the original vector). You can consider a 2D image to be a projection of a bunch of 3D objects onto the plane of the screen.
If the projection of $b$ onto the column space of $A$ is equal to $b$ than you know that $x$ exists, but that seems like a roundabout way to accomplish a straightforward task.
If you want to solve the system, often the easiest and most general way to solve such an equation is through the Row Reduction Algorithm (also known as Gaussian Elimination).
Here is a link explaining the algorithm http://people.math.aau.dk/~ottosen/MMA2011/rralg.html, its a bit tedious and can be difficult to get the hang of but it gets a lot easier with practice.
