I have never liked span questions, so I could use some help with this. First, suppose that is a vector space over a field . Now suppose satisfies . Prove that .
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Take x over to the other side, y+z=-x, to see that (can you see why?) Then, do it the other way!
Saying that v is in the span of x and y means v= ax+ by for some scalars a and b. Since x+ y+ z= 0, x= -y- z so v= a(-y- z)+ by.
Originally Posted by HallsofIvy Saying that v is in the span of x and y means v= ax+ by for some scalars a and b. Since x+ y+ z= 0, x= -y- z so v= a(-y- z)+ by. This means that implies that , right? Or am I misinterpreting something?
Hi ..it also means, and are vectors.
Originally Posted by Runty This means that implies that , right? Or am I misinterpreting something? Well, v= a(-y- z)+ by= (-a+ b)y+ (-a)z so that v is a linear combination of y and z and therefore is in span(y,z). You will also need to prove the other way: if u is in span(y, z), then it is in span(x, y).
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