Question

I do not understand how we can have 4 vectors in R^3? It says show that the vectors v_1=(1,1,1), v_2=(1, 1,0), v_3=(1,0,0), v_4=(3, 2, 0) span R^3 but do not form a basis for R^3?

Answer 1

You don't understand you can have 4 vectors in R^3?

Answer 2

Yeah

Answer 3

well, (1,0,0) is in R^3, so is (2,0,0), (0,1,0) and (121,45,22)

Answer 4

Okay R^3 is like an area and you can have as many vectors as it allows?

Answer 5

like within the range of it right

Answer 6

Yes, there are infinitely many vectors in R^3.

Answer 7

Oh okay so for my question it says show that its a span but do not form a basis

Answer 8

does that mean that the 4 vectors are dependent of each other?

Answer 9

For those vectors to be basis of R^3 they have to be independent.

Answer 10

They don't have to be independent to span R^3.

Answer 11

Yes so I'd show that they are dependent

Answer 12

Yeah I was just wondering how'd you show that it's a span of R^3

Answer 13

I know that if the determinant is non zero then it's a span but it's not a square matrix so I can't do that

Answer 14

Ok, if you add and subtract those vectors from each other, can you see that you can make the vectors (1,0,0), (0,1,0), (0,0,1) and (3,2,0?)

Answer 15

ohh so 0(1,1,1)+2(2,2,0)+2(1,0,0)=(3,2,0)?

Answer 16

oops 0(1,1,1)+2(1,1,0)+2(1,0,0)=(3,2,0)?

Answer 17

Well, yes, you can make (3,2,0) from the first three vectors, but in fact you can make every vector in R^3. That's easiest to see when you have (1,0,0), (0,1,0), (0,0,1).

Answer 18

But I don't have (1,0,0), (0,1,0) or (0,0,1)?

Answer 19

well maybe I do that (1,0,0) but that's it

Answer 20

sorry for not getting this

Answer 21

you can get them by adding and subtracting the vectors. v1-v2=(0,0,1) for example.

Answer 22

Oh I see okay so if I want to make sure something is span of R^n so I check that the vectors make those vectors?

Answer 23

That's right.

Answer 24

okay how about like v_1=(4,3) for R^2?

Answer 25

if you're only given one vector how'd you prove it

Answer 26

You can't span R^2 with only one vector.

Answer 27

see that's the thing I'm slightly confused with b/c you said there's an infinite amount of vectors in R^3?

Answer 28

what happens if you're given only 1 or 2 vectors?

Answer 29

You need at least the same amount of vectors as the dimension of the vectorspace to span that space. So for R^3 you need at least three vectors.

Answer 30

at least 3 but you can have more? oh okay that makes a lot more sense!! thanks for all your help!!

Answer 31

You can have as many as you want to span the space. Not to form a basis though, because they wont be independent.

Answer 32

haha okay so if you have 4 vectors in R^3 it can possibly span but it won't be linearly independent?

Answer 33

That's right.

Answer 34

will it always span? if it's the same amt or greater

Answer 35

No, for example (1,0,0), (2,0,0), (3,0,0), (4,0,0) doesn't span R^3

Answer 36

Oh okay thank you so much, you just taught me something I couldn't get for awhile in like 10 minutes!!

Answer 37

You're welcome.