subspace of the Galerkin projection. The entries of the 2-by-2 Galerkin matrix are , while the components of the right-hand-side vector of the Galerkin equation are . Finally, the approximate solution is obtained from the components of the solution vector of the Galerkin equation and the basis as . Linear equation in a Hilbert space Weak formulation of a linear equation Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space , namely, find such that for all . Here, is a bilinear form (the exact requirements on will be specified later) and is a bounded linear functional on . Galerkin dimension reduction Choose a subspace of dimension n and solve the projected problem: Find such that for all . We call this the Galerkin equation. Notice that the equation has remained unchanged and only the spaces have changed. Reducing the problem to a finite-dimensional vector subspace allows us to numerically compute as a finite linear combination of the basis vectors in . Galerkin orthogonality The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since , we can use as a test vector in the original equation. Subtracting the two, we get the Galerkin orthogonality relation for the error, which is the error between the solution of the original problem, , and the solution of the Galerkin equation, Matrix form of Galerkin's equation Since the aim of Galerkin's method is the production of a linear system of equations, we build its matrix form, which can be used to compute the solution algorithmically. Let be a basis for . Then, it is sufficient to use these in turn for testing the Galerkin equation, i.e.: find such that We expand with respect to this basis, and insert it into the equation above, to obtain This previous equation is actually a linear system of equations , where Symmetry of the matrix Due to the definition of the matrix entries, the matrix of the Galerkin equation is symmetric if and only if the bilinear form is symmetric. Analysis of Galerkin methods Here, we will restrict ourselves to symmetric bilinear forms, that is While this is not really a restriction of Galerkin methods, the application of the standard theory becomes much simpler. Furthermore, a Petrov–Galerkin method may be required in the nonsymmetric case. The analysis of these methods proceeds