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

In summary, the conversation discusses a vector space problem involving the set of entities of the form (a,b,c) with real number entries and defined addition and scalar multiplication. The conversation shows that vectors of the form (a,b,1) do not form a vector space, but does not provide a logical explanation or proof. The conversation also brings up the question of whether the resultant vector would still belong to the set of vectors if two vectors of this form are added.
  • #1
ercagpince
30
0
[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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  • #2
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?
 

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

What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and a set of operations that can be performed on those vectors, such as addition and scalar multiplication. It is a fundamental concept in linear algebra and is used to model and solve various problems in physics, engineering, and computer science.

What is a simple vector space problem?

A simple vector space problem is a problem that involves finding a solution within a vector space using the operations defined on that space. This could include finding linear combinations of vectors, determining if a set of vectors is linearly independent, or solving systems of linear equations.

How do you determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, you can use the definition of linear independence, which states that a set of vectors is linearly independent if and only if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0. In simpler terms, if the only way to get a linear combination of the vectors equal to zero is by setting all the coefficients to zero, then the vectors are linearly independent.

How is a vector space problem solved?

A vector space problem is typically solved by applying the operations defined on the vector space to manipulate the given vectors and equations until a solution is found. This can involve using methods such as Gaussian elimination, matrix operations, and determinants.

What are some real-world applications of vector spaces?

Vector spaces have countless real-world applications, including image and signal processing, machine learning and data analysis, physics and engineering problems, and computer graphics. They are also used in fields like economics, social sciences, and genetics to model and solve various problems.

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