Final answer:
The domain for both sets of transformations (a) and (b) is the set of all ordered triples of real numbers. The codomain for (a) is also the set of all ordered triples of real numbers, while for (b), it is the set of all ordered pairs of real numbers since only two output values are defined.
Step-by-step explanation:
To find the domain and codomain of the given transformations, let's consider each set of equations separately.
For set (a):
w₁= x₁ – 4x₂ + 8x₃, w₂ = -x₁ + 4x₂ + 2x₃, w₃ = – 3x₁ + 2x₂ – 5x₃
The domain of this transformation is the set of all ordered triples (x₁, x₂, x₃) such that each xᵢ belongs to the set of real numbers, ℝ.
The codomain is also the set of all ordered triples of real numbers, as the transformation maps real-numbered inputs to real-numbered outputs.
For set (b):
w₁ = 2x₁ + 7x₂ - 4x₃, w₂ = 4x₁ – 3x₂ + 2x₃
- Similarly, the domain here is the set of all possible ordered triples (x₁, x₂, x₃)
- The codomain, however, is the set of all ordered pairs (w₁, w₂), since only two output values are determined by the transformation.