Solving Matrix-Vector Equations

1. Solving Matrix-Vector Equations

Now that we know some conditions that can ensure that a square matrix-vector equation, or a set of n equations and n unknowns, has a unique solution, we can discuss methods to find such a solution.

2. Solving Matrix-Vector Equations

Solving matrix-vector equations in linear algebra is very similar to solving linear equations in regular algebra. Here 5x equals 7. The multiplicative inverse of 5, 1/5 is multiplied by both sides of the equation to negate the 5 on the left and isolate x, giving us 7/5 as our solution on the right.

3. Solving Matrix-Vector Equations

For matrix-vector equations, it becomes clear why the existence of an inverse is so important, as it serves as the 1/5th in this problem, able to be multiplied on the left of both sides of the equation is isolate the solution vector x. Notice here, just like 1/5 times 5 is 1, A inverse times A is equal to the identity matrix.

4. Solving Matrix-Vector Equations

In R it's as simple as using the solve() command discussed previously, and multiplying the result (should it exist) by the given vector b to obtain x. Thus, to obtain x, the solution to Ax equals b, simply use the solve(A)%*%b command.

5. Solving Matrix-Vector Equations

You can even check your work by multiplying the solution obtained previously on the left by A and see if you obtain b back. Here, the command is A%*%x, which does indeed produce the original b! This helps you know that you did not have a bug in your code!

6. Additional Conditions for Unique Solutions

Since any matrix multiplied by suitably-sized a vector full of zeros produces only a vector full of zeros, it would follow that the homogeneous equation Ax = 0 would have one and only one solution, x = 0, if A is invertible. This is another property that is equivalent to a matrix-vector equation having a unique solution in the case where A is square.

7. Additional Conditions for Unique Solutions

Not to serve as a proof, but notice that, if A is invertible, it creates an inverse matrix, which if multiplied by the zero vector equals zero. Thus, in this case solve(A)%*%b indeed produces a 2-dimensional vector of zeros. The zero vector is called the trivial solution to the homogeneous matrix-vector equation Ax = 0.

8. Conditions for a Unique Solution to Matrix-Vector Equations

Thus, in addition to being invertible, having a non-zero determinant, and having its rows and columns serve as a basis for their vector spaces, having only the trivial solution to Ax = 0 is equivalent to having a unique solution to every equation Ax = b where A is square. The idea is that the only way to produce the zero vector from the columns of the matrix A via a linear combination is to give them all a weight of zero and add them up. There's no other way.

9. Let's Practice!

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