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## Homework Statement

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.

## Homework Equations

## 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.

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