Question

Consider the two-dimensional vector space V = R2 and the ordered basis B = {(1, 2), (4, -15)}. Which of the following is the dual basis of V*?

Answer 1

The dual basis of V*, B* = {f1, f2}, is comprised of linear functionals which correspond to vectors in B, such that each functional maps its corresponding vector to 1, and the other vector to 0. Solving for the linear functionals requires finding the appropriate coefficients that satisfy these conditions.

Step-by-step explanation:

To find the dual basis for the vector space V* which is dual to the vector space V = R2 with respect to the basis B = {(1, 2), (4, -15)}, we need linear functionals that map each vector in B to 1 for itself and 0 for the other vector.

Let B* = {f1, f2} be the dual basis for B. Then we must have:

• f1(1, 2) = 1 and f1(4, -15) = 0,
• f2(1, 2) = 0 and f2(4, -15) = 1.

To find these functionals, we can set f1(x, y) = ax + by and f2(x, y) = cx + dy, where a, b, c, and d are coefficients that we need to determine. By applying the conditions above, we get two systems of linear equations:

1. For f1: a + 2b = 1, 4a - 15b = 0,
2. For f2: c + 2d = 0, 4c - 15d = 1.

Solving these systems will give us the coefficients a, b, c, and d, which define the linear functionals f1 and f2.