Determine whether the set is a basis for the vector space V.V = M22, = 0 1 1 0, 1 1 0 1, −1 1 1 −1

To determine whether a set of vectors forms a basis for a vector space, we need to check two conditions:

1. The vectors are linearly independent.
2. The vectors span the vector space.

The given set of vectors is in M22, the space of 2x2 matrices. The vectors are:

v1 = [0, 1] [1, 0]

v2 = [1, 1] [0, 1]

v3 = [-1, 1] [1, -1]

1. Check for Linear Independence:

We set up the following system of equations to check for linear independence:

av1 + bv2 + c*v3 = 0

This gives us the following system of equations:

0a + 1b - 1c = 0 1a + 1b + 1c = 0 1a + 0b + 1c = 0 0a + 1b - 1c = 0

Solving this system, we find that the only solution is a = b = c = 0. This means the vectors are linearly independent.

2. Check for Span:

In the space of 2x2 matrices, any matrix can be written as a linear combination of the vectors v1, v2, and v3. Therefore, these vectors span the space.

Since the vectors are linearly independent and span the space, they form a basis for the vector space V.