Finding a basis for span of vectors

Question

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

Thanks

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.