Generating the yz-Plane with Vectors
The question is asking to show that the yz-plane, given by the set W = {(0, b, c) | b, c ∈ ℝ}, can be generated by the indicated sets of vectors.
The yz-plane in ℝ³ is the set of all points where the x-coordinate is 0. The general point in this plane can be written as (0, y, z), where y and z can take any real values.
(i) To show that W is generated by the vectors (0, 1, 1) and (0, 2, -1), we need to express any vector (0, b, c) in W as a linear combination of (0, 1, 1) and (0, 2, -1).
Let's try to express any point (0, b, c) as a linear combination of (0, 1, 1) and (0, 2, -1):
(0, b, c) = α(0, 1, 1) + β(0, 2, -1)
Expanding this, we have:
(0, b, c) = (0, α + 2β, α - β)
We want to solve for α and β such that the second and third components of the vectors match. This gives us two equations:
α + 2β = b
α - β = c
We can solve these equations simultaneously to find α and β in terms of b and c. Adding the two equations, we get:
2α + β = b + c
Subtracting the second equation from the first one, we get:
3β = b - c
Thus, β = (b - c) / 3. Substitute β back into one of the original equations to get α. For example, using the first equation:
α = b - 2β
α = b - 2(b - c) / 3
α = (3b - 2b + 2c) / 3
α = (b + 2c) / 3
So any vector (0, b, c) can be represented as a linear combination of (0, 1, 1) and (0, 2, -1), with the coefficients α = (b + 2c) / 3 and β = (b - c) / 3.
(ii) Similarly, to show that W is generated by the vectors (0, 1, 2), (0, 2, 3) and (0, 3, 1), we need to express any vector (0, b, c) in W as a linear combination of these vectors.
(0, b, c) = α(0, 1, 2) + β(0, 2, 3) + γ(0, 3, 1)
Expanding this, we have:
(0, b, c) = (0, α + 2β + 3γ, 2α + 3β + γ)
Again, set up a system of equations to solve for α, β, and γ:
α + 2β + 3γ = b
2α + 3β + γ = c
This system has two equations with three unknowns, making it underdetermined. However, since the third component isn't given (as there is no x-component in the vectors generating W), we can freely choose γ (for example, γ = 0), and the other two can be determined from the equations above. This way, every vector in the yz-plane can be expressed as a combination of these three vectors.
For any given b and c, we can always find α and β (and choose γ) such that (0, b, c) is a linear combination of (0, 1, 2), (0, 2, 3), and (0, 3, 1), confirming that these vectors generate the yz-plane.
