Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$U = \text{span}\{(1,0,0),(0,2,-1)\}$, $W = \text{span}\{(0,1,-1)\}$. 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 $W$?


share|cite|improve this question
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

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.