Determine whether the vectors (1, 0), (0, 1), and (1, 1) form a basis for R^2.

Step 1: Recall the requirements for a basis.

A basis for R^2 must satisfy two conditions:
1. The vectors must span R^2.
2. The vectors must be linearly independent.

A basis for R^2 contains exactly 2 linearly independent vectors.

Step 2: Examine the given vectors.

The vectors are:
v1 = (1, 0)
v2 = (0, 1)
v3 = (1, 1)

Step 3: Check for linear dependence.

Notice that:
v1 + v2 = (1, 0) + (0, 1)

v1 + v2 = (1, 1)

Therefore:
v3 = v1 + v2

Step 4: Interpret this relationship.

Because one vector can be written as a linear combination of the others, the three vectors are linearly dependent.

Step 5: State the conclusion.

Although these vectors span R^2, the set is not linearly independent.

Therefore, the three vectors do not form a basis for R^2.

However, the subset {(1, 0), (0, 1)} does form a basis for R^2.