Start by checking if are linearly independent. We can do this by setting up the following Augmented Matrix:
Now we perform Gaussian Elimination to arrive at the following:
From this we can see that:
So we can see that are not linearly independent. If we set then we get the following:
From this we can see that can be written as a linear combination of . So This means that satisfy the first part of being a Basis now we need to see if they satisfy the second part by looking back at our augmented matrix without column 4:
We can easily see here that non of the columns can be expressed by any other combination of the other columns so they must be linearly independent satisfying our second condition for a basis. This means that are the basis of .
Since we know the basis of it is very easy to find since we know all the vectors in a basis of so since there were three we can say: .