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Homework Help: Lin. Alg. Subspaces Problem

  1. Apr 7, 2013 #1
    1. The problem statement, all variables and given/known data Let W be the set of all ordereed pairs of real numbers, and consider the following addition and scalar multiplication operations on U=(u1,u2) and V=(v1,v2)

    U+V is standard addition but kU=(0, ku2)

    2. Relevant equations Is W closed under scalar multiplication?

    3. The attempt at a solution I understand that W is not a vector space but my book suggests that it is a subspace closed by scalar multiplication.

    Is it because kU=(k0, ku2) where k multiplies both terms?
  2. jcsd
  3. Apr 8, 2013 #2


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    Being closed under scalar multiplication just means that if [itex]w \in W[/itex] and [itex]k \in \mathbb{R}[/itex] then [itex]k w \in W[/itex].
    Since W doesn't really have any restrictions, you only need to check that (0, ku2) is an ordered pair of real numbers.
  4. Apr 8, 2013 #3
    Is [tex] kU \, = \, (0, ku_{2})[/tex] an ordered pair of reals ? Also, does W contain the zero vector ?
  5. Apr 8, 2013 #4


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    It's pretty strange to talk about subspaces here, since W isn't defined as a subset of a vector space. It's just defined as a set with two operations which may or may not turn it into a vector space.

    If it turns out to be closed under both addition and (this non-standard) scalar multiplication, then you can check if it satisfies the eight vector space axioms, to see if it's a vector space.
  6. Apr 8, 2013 #5
    Thanks for the help! That solves it for me completely! I thoroughly didn't understand the definitions of subspace and what they mean and represent, so this simple explanation really clears up a big misconception that I had.
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