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

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Homework Help Overview

The problem involves finding two vectors u and v such that their sum equals a given vector z in R^6, with specific constraints on the coordinates of u and v. The original poster expresses uncertainty about how to approach the problem, particularly in relation to the concepts of direct sums and vector addition.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to understand the nature of the problem and questions whether it relates to direct sums. They also seek clarification on the implications of u's coordinates being equal and the requirement for v's coordinates to sum to zero. Other participants engage by asking for clarification on the constraints and suggesting a decomposition of the vector z into components.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the formulation of a system of equations based on the decomposition of z, but there is no explicit consensus on the approach yet.

Contextual Notes

The original poster notes that they are currently studying direct sums, which may influence their understanding of the problem. There is also a mention of a potential system of equations that could arise from the constraints provided.

deadlycrayon
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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?
 
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deadlycrayon said:
u has coordinates equal to 0?

What do you mean with this?
 
micromass said:
What do you mean with this?

oops sorry i mean the coordinates are all equal
 
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).
 

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