- Suppose we have equations:
- When you have a system where every equation is equal to zero on the right, it is called a homogeneous system
- When you have non-zero values of constants on the side, it is called an inhomogeneous system
- If you inhomogeneous system has a solution, it has as many solutions as its corresponding homogenous system
- This relates to Null Spaces, since every solution of the kernel will be equal to zero
where is a real numbers, and w is the homogenous solutions solves the inhomogeneous
- We can represent the problem as
Example: Solving a system of linear equations
We can write this as a matrix as . Eventually, we want to get to . Alternatively, we can write this as an augmented matrix: We want to perform row operations on this to get to our desired state. We can interchange rows, multiply rows by non-zero scalars, and subtract and add rows from others. We can show that these operations are also linear isomorphisms. We can set row 2 to row 1 plus row 2, and row one as row 1 minus row 3. We can set row 2 as row 2 minus 3/2 times row 3 We can write row 3 as row 3 minus 2/3 row 2 We can write row 1 and row 1 minus 4/3 times row 2 Finally, we can scale to get 1s Interchange the rows to get And we get our solution here.
Example: Unsolvable system
We can write this as . r’1 = r1 - r2 r’2 = r2 - r3 r’1 = r1 - r3 r’4 = r4 -r1 Since the last row has all zeros, we know that this system is not solvable, unless , in which it might be.
Example: Row reduction as a matrix multiplication. Suppose we have a matrix
. Let r’2 = r2 - 5/2 r1. Our matrix becomes . Doing this row operations is the same as left multiplication by this matrix: We can see this as .
Example: Computing an inverse of a matrix
. We want to do row operations such that we get an identity matrix on our left, so we get out inverse on the right.