PROVE THAT PRINCIPLE OF SUPERPOSITION HOLDS IN HOMOGENOUS LINEAR EQUATION

Sure, here is a step-by-step proof that the principle of superposition holds in homogeneous linear equations:

1. Let's start with the definition of a homogeneous linear equation. A linear equation is said to be homogeneous if it can be written in the form Ax = 0, where A is a matrix and x is a vector.

2. The principle of superposition states that the sum of two solutions to a linear homogeneous equation is also a solution.

3. Let's assume that x1 and x2 are solutions to the homogeneous equation Ax = 0. This means that Ax1 = 0 and Ax2 = 0.

4. Now, let's consider the sum of x1 and x2, which we'll call x3. So, x3 = x1 + x2.

5. We substitute x3 into the equation Ax = 0 to get A(x1 + x2) = 0.

6. By the distributive property of matrix multiplication, this simplifies to Ax1 + Ax2 = 0.

7. But we know from our initial assumption that Ax1 = 0 and Ax2 = 0. So, the equation simplifies to 0 + 0 = 0, which is obviously true.

8. Therefore, x3 = x1 + x2 is also a solution to the equation Ax = 0.

9. This proves that the principle of superposition holds for homogeneous linear equations.