Vectors: Given z in u+v=z, find u and v (with constraints)

[deadlycrayon]

Homework Statement

Given a vector z=<-12, 1, 1, 2, 7, 0> in R^6
and z=u+v, then find u and v such that u's coordinates are all equal to each other (like <0,0,0,0,0,0>) and v has coordinates that add up to 0

Homework Equations

z=u+v

The Attempt at a Solution

i have no idea how to approach this...is this a problem about direct sums? (since that is what we're studying at the moment) On a related note, what is the difference between regular addition and a direct sum?

 

[micromass]

u has coordinates equal to 0?

What do you mean with this?

 

[deadlycrayon]

What do you mean with this?

oops sorry i mean the coordinates are all equal

 

[micromass]

Allright, so you want to make the following decomposition:

$$(-12, 1, 1, 2, 7, 0)=(a,a,a,a,a,a)+(b,c,d,e,f,g)$$

where b+c+d+e+f+g=0.

Now, can you derive a system of equations from this? I claim that you can obtain a system of 7 equations and 7 indeterminates (which are of course a,b,c,d,e,f,g).