Let U be the span of
[ 0 ] [ 2 ]
[ 1 ] and [ 0 ]
[ 2 ] [ -1]
[ 4 ] [ 1 ]
Give conditions on a, b, c and d so that [a b c d] (transposed) is in U (as in, give restrictions that are equations of only a, b, c and d.
The Attempt at a Solution
I've set up the problem to set the two given vectors multiplied by scalar x and y, yielding four equations:
2y = a,
x = b,
2x - y = c, and
4x + y = d.
Is this the right first step to take? After that, I take x or b as a free variable, and try to rewrite the other equations in terms of b, yielding:
2y = a, b = b, 2b - y = c, and 4b + y = d.
After this, however, I am stuck. Any pointers? Did I set up the problem the right way?
Edit: I also know that [a b c d] must be a linear combination of the two given vectors if it is in the span. However, if it is a linearly independent set, the trivial zero vector could be matched with [a b c d], although I am pretty sure the answer isn't that simple.