Final answer:
To find a basis for the set of vectors in R³ in the plane x + 2y + z = 0, we need to find two linearly independent vectors that lie in the plane. By setting z = 0, we can solve for x and y to obtain two linearly independent vectors (-2, 1, 0) and (2, -1, 0). Therefore, a basis for the set of vectors in R³ in the plane x + 2y + z = 0 is {(-2, 1, 0), (2, -1, 0)}.
Step-by-step explanation:
To find a basis for the set of vectors in R³ in the plane x + 2y + z = 0, we need to find two linearly independent vectors that lie in the plane. We can do this by setting z = 0 and solving for x and y. By setting z = 0, we get x + 2y = 0, which can be rewritten as y = -x/2. Now, we can choose two values for x, say x = 2 and x = -2, and find the corresponding values of y. For x = 2, y = -2/2 = -1, and for x = -2, y = -(-2)/2 = 1. Therefore, we have two vectors (-2, 1, 0) and (2, -1, 0) that are linearly independent and lie in the plane. Hence, a basis for the set of vectors in R³ in the plane x + 2y + z = 0 is {(-2, 1, 0), (2, -1, 0)}.