Projections in field R of real numbers with N dimensional vector space

transpose(A)(b-Ax) = 0 (The zero vector because b-Ax is a vector)

The above equals:

transpose(A)(b) – transpose(A)Ax =0

transpose(A)(b) = transpose(A)Ax

If the matrix is full column rank or of course full rank we can multiply the left inverse of transpose(A)A and get the identity matrix.

(transpose(A)A)¯¹*transpose(A)Ax   =  (transpose(A)A)¯¹*transpose(A)(b)

Which leaves us with the identity matrix on the left side.

x=  (transpose(A)A)¯¹*transpose(A)(b)