What does it actually mean to solve the system Mx = b?
For some students, it's entirely likely that after doing all the math in
your linear algebra course, you forgot the original reason why you defined
these matrices, and why you're learning Gaussian elimination.
For the one dimensional case, one solves mx = b for
x. In this case, the solution is simple: x = b/m.
Let's write out the matrix vector equaiton Mx = b. Given the matrix
and the vectors x = (x1, x2)T and b = (b1, b2)T, then expanding we get:
It's important here to remember that all the values mi, j and
bi are given, a concrete example, is useful.
Example of a 2×2 System of Linear Equations
What does Mx = b mean when
All this is, in disguise, is
4x1 + 2x2 = 3
3x1 + 5x2 = 4
which is equivalent to
4x1 + 2x2 − 3 = 0
3x1 + 5x2 − 4 = 0
However, note that the left hand side describes a plane. In Figure
1, we show f1(x) = 4x1 + 2x2 − 3,
and in Figure 2, we show
f2(x) = 3x1 + 5x2 − 4.
Figure 1. The function f1(x) = 4x1 + 2x2 − 3.
Figure 2. The function f2(x) = 3x1 + 5x2 − 4.
The first function, f1(x) is zero on the line
x1 = 3/4 − 2/4x2, while the second
function, f2(x) is zero on the line
x1 = 4/3 − 5/3x2. What it means to solve a linear
system is that we are finding a value of x = (x1, x2)T such
that both f1(x) = 0 and f2(x) = 0.
In this example, there is one unqiue point, and that point may be seen in Figure 3.
Figure 3. Both function f1(x) and f2(x).
These two functions are simultaneously zero at only one point, seen in Figure 3
as the point where the grey (z = 0), red, and blue planes intersect. From
the linear algebra you learned, you could solve this system to find that the
solution is x = (0.5, 0.5)T.
Thus, solving a system of n linear equations is no more than finding
simultaneous roots of n different linear functions. This is why we can use
such techniques as when we perform Newton's method in n dimensions: we
convert a system of n nonlinear functions into a simplified problem with
n linear functions, find the root of the linear functions and show that, under
the appropriate conditions, the solution to the linear functions is a good approximation
to the nonlinear problem.
Why are we Solving Systems of Linear Equations?
The most obvious example comes in examples such as trying to solve circuit in
Figure 4. Kirchhoff's voltage laws tells us that the sum of the voltages in each
of the loops is zero.
Figure 4. A simple circuit.
In the first loop, we get 7 V − 1 (i1 + i2) − 2 i1 = 0, and in the
second, we get 2 i1 − 4 i2 = 0.
Solving these yeilds the values i1 = 2 A and i2 = 1 A.