Let w, x, y, z be vectors and suppose z = 2x + y and w = 4x - y - 2z.

Question

Let w, x, y, z be vectors and suppose z = 2x + y and w = 4x - y - 2z.

Mark the statements below that must be true.

A. Span(x, y)=Span(w)

B. Span(y, w)= Span(z)

C. Span(y) = Span(w)

D. Span(x, y) = Span(x, w, z)

Answer 1

This is an exercise in recognizing the relationship between linear independence and span. For example, if two vectors a and b are linearly dependent, then Span(a, b) = Span(a), or if c can be written as a linear combination of a and b, then Span(a, b, c) = Span(a, b).

A. Can x or y be written solely in terms of w? (You may find that x cannot necessarily).

B. Can y be written solely in terms of z? (You may find that it cannot necessarily)

C. Try substituting z for 2x+y in the expression for w and see what happens.

D. After part C, this becomes simpler, because then you find that x, w, z are linearly dependent because z is a linear combination of x and y.