Question

span {u,u-v} contains the vector v. True or False?

Answer 1

If something is in the span of a set of vectors, that just means that we can combine these vectors by adding up multiples of them to get to that vector.

So can we multiply the vectors u and (u-v) by scalars and add them up to reach the vector v?

Answer 2

\[
a\mathbf u + b(\mathbf u-\mathbf v) = a\mathbf u + b\mathbf u-b\mathbf v =(a+b)\mathbf u-b\mathbf v
\]

Answer 3

So we want \[\begin{array}{}
-b &=& 1 \\
a+b &=& 0
\end{array}
\]

Answer 4

If this system has a solution, the \(\mathbf v\) is in the span.

Answer 5

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