spanning set

Quote:

Originally Posted by alexandrabel90

how do you find the spanning set for U ∩ V where U={(x,y.0): x and y are complex} and V= sp { (1,2,3,), (i,-i, 10)}?

Clearly: U contains only vectors from V that have z-coordinate 0, thus if you take any linear combination $\lambda (1,2,3)+\mu(i,-i,10)$ from V, you have to require that $\lambda \cdot 3 +\mu\cdot 10=0$ , hence $\lambda=-\frac{10}{3}\mu$ .
This means that $U\cap V$ is 1-dimensional and that any single vector that you get by chosing $\mu,\lambda \neq 0$ and $\lambda=-\frac{10}{3}\mu$ will therefore span that intersection.
Take for example, $\mu=3$ , hence $\lambda=-\frac{10}{3}\cdot 3 = -10$ . This gives the vector
$-10(1,2,3)+3(i,-i,10)=(-10+3i,-20-3i,0)$ of $U\cap V$ .