Final answer:
With the provided vectors u1 and x, we cannot form an orthogonal basis for R² since the second basis vector u2 is missing, nor can we express x as a linear combination of u1 alone as the equations for the scalar multiple are inconsistent.
Step-by-step explanation:
The question asks to show that {u1, u2} or {u1, u2, u3} is an orthogonal basis for R² or R³, respectively, and then express x as a linear combination of the u vectors. Given vectors u1 = [2, -3], no u2 or u3 is provided, and x = [9, -7], we must check if u1 and x are orthogonal in R². However, with the given information, we can't form a complete basis or check orthogonality as the second vector, u2, is missing, and there's no third vector, u3, to form a basis for R³. Despite this, we can still express x as a linear combination of u1 alone (although not as an orthogonal basis).
Expressing x as a linear combination of u1, we can solve a system of equations to find the scalar multiple a such that x = a * u1. The system is:
2a = 9
-3a = -7
Solving this, we find that the scalar multiple a does not exist as the two equations give different values for a, indicating that x cannot be expressed as a multiple of u1 alone. Thus, more information (like vector u2) is needed to properly address the question of orthogonality and expressing x as a linear combination of the basis vectors.