Expanding linear independent vectors

[member 428835]

Hi PF!

The other day in class my professor mentioned something about expanding linear independent vectors, but he did not elaborate. From what I understand, if $x_1,...,x_k$ are linearly independent vectors in $V$, where $dimV=n>k$, how would you extend $x_1,...x_k$ to a basis $\{ x_1,...,x_n \}$. Let's say $\{ y_1,...,y_n \}$ is a basis for $V$. By extending the $x$ vectors, do you think he was just referring to including all the $y$ vectors in the set of $x$ vectors that are linearly independent of the $x$ vectors?

Thanks!

 

[PeroK]

If you have a set of linearly independent vectors, you can expand that set into a basis simply be adding more vectors. In your example, you can choose $x_{k+1}$ as any vector in V that is not in the span of $\{x_1, \dots x_k \}$.

And then, any vector $x_{k+2}$ that is not in the span of $\{x_1, \dots x_{k+1} \}$.

Until you have a basis of $n$ linearly independent vectors.

 

[member 428835]

OK cool, that's what I thought but I wanted someone else's perspective! Thanks PeroK!