Let w x y z be vectors and suppose z 4x y and w 8x 4y 2

Let w, x, y, z be vectors and suppose z = -4x + y and w = 8x + 4y + 2z. Mark the statements below that must be true. Span (y, w)= Span(z) Span(x, z)= Span(y, w) Span(x, y) = Span(x, w, z) Span(y) = Span(w)

Solution

z can be written in terms of x and y

So ,w=8x+4y+2z=8x+4y+2(-4x+y)=6y

w=6y or y=w/6

z=-4x+y=-4x+w/6 or x=(-z+w/6)/4

HEnce,

So, z and w can be written in terms of x and y

And Span(y)=Span(w) as y is a multiple of w

C is also true.