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!