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Vectors and linear dependence

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?
Reply 1
Avatar for Gorrilaz
Gorrilaz
OP
1
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?
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. R4\mathbb{R}^4 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 (R3\mathbb{R}^3 in this case)

Hint: since R3\mathbb{R}^3 is a three dimensional space, there will be only three vectors in the basis
(edited 11 years ago)
Reply 3
Avatar for ztibor
ztibor
10
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
v⃗=α⋅v⃗1+β⋅v⃗2+γ⋅v⃗3\vec v=\alpha \cdot \vec v_1+\beta \cdot \vec v_2 +\gamma \cdot \vec v_3
It means 3 scalar equations . Solve these simoultaneously for α,β,γ\alpha, \beta,\gamma

Another method would be the base transformation.

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