Calculating Projection Matrices

This example goes over finding the Projection Matrix and the Orthogonal Projection Matrix of a given subspace .

Problem

Calculate the projection matrices of and where:
Note has an orthogonal basis.

Solution

There are two ways to approach this problem:

Approach 1:
Start by finding using the formula:
Then we can find using the following:

Approach 2:
Start by finding :
Now we can find using the following formula:
Then we can find using the following:

From this example we see that we want to choose the or based on their dimension then we use the formula to find the other projection matrix.