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Math Help - spanning set

  1. #1
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    spanning set

    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)}?
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    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.
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