Question

Find a basis for the space spanned by the given vectors:

[2, 7, -3, -6, 0],
[3, 3, 2, -3, 3],
[1, -2, 1, 2, 2].

Answer 1

To find a basis for the space spanned by the given vectors, set up a system of equations and solve for the variables. If the system has a unique solution, the vectors are linearly independent and form a basis.

Step-by-step explanation:

To find a basis for the space spanned by the given vectors, we need to determine if the vectors are linearly independent. We can do this by setting up a system of equations and solving for the variables. If the system has a unique solution, then the vectors are linearly independent and form a basis for the space spanned by them.

Let's set up the augmented matrix for the system:

[2 7 -3 -6 0] [3 3 2 -3 3] [1 -2 1 2 2 | 0]

Then, bring it to reduced row-echelon form:

[1 0 0 1 1 5] [0 1 0 -1 2 1] [0 0 1 -2 -3 -4]

So, the system has a unique solution, which means the vectors are linearly independent. Therefore, a basis for the space spanned by the given vectors is {[2, 7, -3, -6, 0], [3, 3, 2, -3, 3], [1, -2, 1, 2, 2]}.