I keep hearing that if you have an RR^3 space you need 3 elements to fill the space. So, let's consider a column vector. I understood 3 elements to mean = 3 column vectors with 3 elements each. As you can't span RR^3 with JUST one or two vectors even if they each have 3 elements. Correct? Could you technically somehow take 2 column vectors with 4 elements to span RR^3? I'm guessing not.

2k1ablakrh0 2022-09-17 Answered
I keep hearing that if you have an R 3 space you need 3 elements to fill the space.
So, let's consider a column vector. I understood 3 elements to mean = 3 column vectors with 3 elements each. As you can't span R 3 with JUST one or two vectors even if they each have 3 elements. Correct?
Could you technically somehow take 2 column vectors with 4 elements to span R 3 ? I'm guessing not.
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Answers (1)

unfideneigreewl
Answered 2022-09-18 Author has 5 answers
To the first question. You need three vectors v 1 , v 2 , v 3 R 3 . It's correct. The elements refers to vectors v R 3 . But that is not enough to span R 3 , they have to be linearly independent; i.e.
a 1 v 1 + a 2 v 2 + a 3 v 3 = 0  iff  a 1 = a 2 = a 3 = 0.
The second question. No. First because a vector with 4 entries lives in R 4 . But if you consider R 3 embedded in R 4 . Again you need 3 vectors v 1 , v 2 , v 3 R 4 that are linearly independent to span R 3 , but here the question is a little bit different. Think of R embedded in R 2 . You can embed it as any line that passes through the origin, so you have many representations of R in R 2 , same thing happens with R 3 embedded in R 4

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