Linear Independence. express each vector as a lin. combo

[burton95]

V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R⁴

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this

 

[Ray Vickson]

V1 = (1,2,3,4) V2 = (0,1,0,-1) V3 = (1,3,3,3)

a) I already expressed them a linearly dependent set in R⁴

b) Express each vector in part (a) as a linear combination of the other two

linear combo is just {c1v1 + c2v2...cnvn} right? But I don't get where to start to prove this

There is nothing to prove; that is just a *definition* of "linear combination". You are being asked to represent v1 as a linear combination of v2, v3, and so forth. To start: just write things down in detail: figure out what are the components of c2v2 + c3v3 for constants c2 and c3. How can that combination be equal to v1?

 

[burton95]

so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?

 

[Mark44]

What Ray is saying has three parts.
1) Find constants c₂ and c₃ for which v₁ = c₂v₂ + c₃v₃
2) Find constants c₁ and c₃ for which v₂ = c₁v₁ + c₃v₃
3) Find constants c₁ and c₂ for which v₃ = c₁v₁ + c₂v₂

 

[Mark44]

BTW, when you post a question, do not throw away the three parts of the template. They are there for a reason.

 

[burton95]

Thx. I will leave the template. So I am left with 3 different equations in the form of vn = cx(vx) + cy(vy) as my final answer?

 

[Ray Vickson]

so are you saying take v1 = c2 (v2) + c3 (v3) using the constants that I found through proving that they are linearly dependent?

That would be one way of doing it.