Question

Say you have a set of linearly independent vectors {v1,v2,v3} that span R^3. Is it safe to say that the set {v1,v2} is also linearly independent? Does it also span R^3?

Answer 1

are u gonna give urself a medal lol

Answer 2

In short, It's definitely still linearly independent, but it doesn't span R^3.

Answer 3

How so? It's the same vectors that span R^3? Nothing has changed for the vectors?

Answer 4

For example, since {v1, v2, v3} are linearly independent, there is no combination of {v1, v2} that results in v3. Thus, we're missing at least one element in R^3, so it can't span R^3.

Answer 5

Alternatively, you know that {v1, v2, v3} is a basis for R^3, and every basis for R^3 must have at least 3 vectors. Since {v1, v2} only has 2 vectors, it can't span R^3.

Answer 6

I should have said "has exactly 3 vectors"

Answer 7

Ok so if the first set was lin dependent and the second was also lin dependent then the second set would also span R^3 right?

Answer 8

If the first set were lin. dependent, then it wouldn't span R^3. Same thing for the second set.

Answer 9

Ok I'm starting to get it now.

Answer 10

Ok so if you have 3 vectors and 2 of those vectors are a linear combination of each other (linear dependent) then the set will only span R^2
@KingGeorge

Answer 11

Yes.

Answer 12

Ok I get how span relates to lin dependence and lin independence now. Thank you.

Answer 13

You're welcome.