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Vectors and linear dependence

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1. Show vectors (0,3,1-1) (6,0,5,1) (4,-7,1,3) form a linearly dependent set in "R^4"

Does this just mean show the vectors are linearly dependent as I know how to do that, its just the R^4 thats confusing me.

Also which of the following vectors form a basis for R^3?
(1,0,0) (2,2,0) (3,3,3)

(3,1,-4) (2,5,6) (1,4,8)

Is this again just asking which of the vectors are a linearly dependent? I would probably know how to work it out, I just can't get my head around the wording of the question, so I'm not sure what they are asking?
2. Also find the vector v for the basis S=(v1,v2,v3)

v=(2,-1,3)
v1=(1,0,0)
v2=(2,2,0)
v3=(3,3,3)

Any help on to approach this question would be appreciated, im not really sure what Im meant to be looking for here. Isn't the vector v already given?
3. (Original post by Gorrilaz)
Show vectors (0,3,1-1) (6,0,5,1) (4,-7,1,3) form a linearly dependent set in "R^4"

Does this just mean show the vectors are linearly dependent as I know how to do that, its just the R^4 thats confusing me.

Yes, you only need to show the vectors are linearly dependent. is simply the vector space concerned.

(Original post by Gorrilaz)
Also which of the following vectors form a basis for R^3?
(1,0,0) (2,2,0) (3,3,3)

(3,1,-4) (2,5,6) (1,4,8)

Is this again just asking which of the vectors are a linearly dependent? I would probably know how to work it out, I just can't get my head around the wording of the question, so I'm not sure what they are asking?
No. To show that vectors from a basis for a space, you have to show (a) that they form a linearly independent set and (b) that they span the vector space ( in this case)

Hint: since is a three dimensional space, there will be only three vectors in the basis
4. (Original post by Gorrilaz)
Also find the vector v for the basis S=(v1,v2,v3)

v=(2,-1,3)
v1=(1,0,0)
v2=(2,2,0)
v3=(3,3,3)

Any help on to approach this question would be appreciated, im not really sure what Im meant to be looking for here. Isn't the vector v already given?
If (v1, v2, v3 ) is a basis then

It means 3 scalar equations . Solve these simoultaneously for

Another method would be the base transformation.

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