Solving a Vector Space Problem: (a,b,1) Not a Vector Space

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SUMMARY

The discussion addresses the problem of proving that vectors of the form (a,b,1) do not constitute a vector space under standard vector operations. The key conclusion is that the z-component of the resultant vector from the addition of two vectors of this form does not remain constant at 1, violating the closure property required for vector spaces. The participant successfully identified the inverse and null vector but struggled to articulate the proof of non-closure for this specific set.

PREREQUISITES
  • Understanding of vector space properties, including closure, associativity, and identity elements.
  • Familiarity with vector addition and scalar multiplication definitions.
  • Knowledge of real number operations and their implications in vector spaces.
  • Ability to construct mathematical proofs and logical arguments.
NEXT STEPS
  • Study the properties of vector spaces in linear algebra.
  • Learn about closure properties and their importance in defining vector spaces.
  • Explore examples of sets that do and do not form vector spaces.
  • Practice constructing proofs for various vector space problems.
USEFUL FOR

Students studying linear algebra, educators teaching vector space concepts, and anyone interested in mathematical proofs related to vector spaces.

ercagpince
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[SOLVED] a simple vector space problem

Homework Statement



Consider the set of all entities of the form (a,b,c) where the entries are real numbers . Addition and scalar multiplication are defined as follows :
(a,b,c) + (d,e,f) = (a+d,b+e,c+f)
z*(a,b,c) = (za,zb,zc)

Show that vectors of the form (a,b,1) do not form a vector space .

Homework Equations



all equations defining a vector space

The Attempt at a Solution



I managed to find the inverse under addition vector and also the null vector for that vector space , however , I couldn't find any logical explanation or proof why a vector like (a,b,1) do not form a vector space .
 
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ercagpince said:
Show that vectors of the form (a,b,1) do not form a vector space .

What is the z-component of the resultant vector if you add two of these? Will it still belong to that set of vectors?
 

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