We have z = -3x - 2y and w = 6x + 3y - 2z. We will use these expressions to determine which of the given statements are true.
A. Span(y) = Span(w)
False. Since w is a linear combination of x, y, and z, and z is a linear combination of x and y, we can write w as a linear combination of x and y. Therefore, Span(w) is a subset of Span(x, y), but it is not necessarily equal to Span(y).
B. Span(x, y) = Span(w)
True. We can rewrite w as:
w = 6x + 3y - 2z
w = 6x + 3y - 2(-3x - 2y)
w = 12x - 3y
Therefore, Span(w) is a subset of Span(x, y), and Span(x, y) is a subset of Span(w), so they are equal.
C. Span(y,w) = Span(z)
True. We can rewrite z as:
z = -3x - 2y
z = -3x - 2y + w - 6x - 3y
z = -9x - 5y + w
Therefore, Span(z) is a subset of Span(y, w), and Span(y, w) is a subset of Span(z), so they are equal.
D. Span(x, y) = Span(x, w, z)
False. Since w is a linear combination of x, y, and z, Span(x, w, z) is a subset of Span(x, y). However, z is not a linear combination of x and y, so Span(x, y) is not a subset of Span(x, w, z). Therefore, the two spans are not necessarily equal.