Find Shortest Distance

This example goes over finding the shortest distance between a vector and a subspace .

Problem:

Find the shortest distance from to where:

Solution

There are two options to solve this problem due to the Projection Theorem:

Option 1:
Find , then , but ensure that has an orthogonal basis, if not you may need to use the Gram-Schmidt Method.

Option 2:
Find , then .

In our case the dimension of of is less than so we will continue with option 2.

We know that where has 's basis as it's rows. So we get that:
Now we find the null-space of is:
Now all we are left to do is find .