Knowing if spans overlap

Question

Only the first checked squares are deemed to be correct. Why is D not correct? After all, the vectors do overlap on the same plane...

Answer 1

Let's explicitly show that all are true as long as the vector space is over a field. Note that $w$ and $z$ are both in Span($x,y$).

1. As $w = 25x + 25y$, we see that $w - 25x = 25y$, and so $y$ is in Span($w,x$). So for A, both sides are equal to Span($x,y$).

2. As $z$ is in Span($x,y$), both sides of B are equal to Span($x,y$).

3. As $z - 6x = 5y$, we see that $y$ is in Span($x,z$). So in fact Span($x,z$) = Span($x,y$). And as $5z - w = 5x$, we have firstly that $x$ is in Span($w,z$). But once we have $x$, we also get $y$, just as in 1. So Span($x,y$) = Span($w,z$).

4. We already know that Span($x,y,z$) = Span($x,y$). As $w - 25x = 25y$, we see that $y$ is in Span($x,w$). So again, both sides are Span($x,y$).

What you have are eight ways of writing Span($x,y$).