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$U = \text{span}\{(1,0,0),(0,2,-1)\}$, $W = \text{span}\{(0,1,-1)\}$. How can I find bases for $U$ and $W$? (I think they're linearly independent, right?)

Can I just take $B_1 = \{(1,0,0),(0,2,-1)\}$ for $U$, and $B_2 = \{(0,1,-1)\}$ for $W$?


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Yes! The elements that span a space are basis elements if they're linearly independent. This is the case here, so your answer is correct. – Marc Apr 5 '14 at 10:05
Yes, you are correct. In general, to find a basis from a spanning set one completes the sifting algorithm (see page 8). – ah11950 Apr 5 '14 at 10:07

A basis $B$ of a vector space $V$ satisfies two properties:

  • $B$ is linearly independent, and
  • $\mathrm{span}(B)=V$.

In this case, we have $U=\mathrm{span}\{(1,0,0),(0,2,-1)\}$.

To verify that $B:=\{(1,0,0),(0,2,-1)\}$ is a basis for $U$, we need to check

  1. $B$ is linearly independent. This is obvious by inspection -- the only solution to $a(1,0,0)+b(0,2,-1)=(0,0,0)$ is the trivial one.

  2. $\mathrm{span}(B)=U$. This is true by definition.

The case of $W$ is even easier.

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