Why do we try to find if a subset is a subspace of a vector space?

[musicgold]

Homework Statement

This is not a homework problem. I am studying abstract algebra on my own and have a question.

I know how to verify if a subset is a subspace of a vector space using the three conditions. What I am trying to imagine is a situation when one will need to verify if a subset is a subspace of a vector space.

Relevant Equations

Consider a set V consist of vectors representing houses in a town, where every house is represented by a column vector with three elements :
v = ( price in $K, sq. foot, # of bathrooms )

I am assuming the set $V$ will have elements like the ones shown below.
$v_{1} = (200, 700, 2)$
$v_{2} = (250, 800, 3)$
...

1. What will be the vector space in this situation?
2. Would a subspace mean a subset of V with three or more bathrooms?

 

[Mark44]

Homework Statement: This is not a homework problem. I am studying abstract algebra on my own and have a question.

I know how to verify if a subset is a subspace of a vector space using the three conditions. What I am trying to imagine is a situation when one will need to verify if a subset is a subspace of a vector space.
Homework Equations: Consider a set V consist of vectors representing houses in a town, where every house is represented by a column vector with three elements :
v = ( price in $K, sq. foot, # of bathrooms )

I am assuming the set $V$ will have elements like the ones shown below.
$v_{1} = (200, 700, 2)$
$v_{2} = (250, 800, 3)$
...

1. What will be the vector space in this situation?
2. Would a subspace mean a subset of V with three or more bathrooms?

This doesn't make much sense as a vector space, since it's meaningless to add two vectors or to multiply either by a scalar.

If you are given an actual vector space V, you know it must satisfy all of the axioms of a vector space. If you don't know what they are, do a web search for "vector space axioms."

To check that a set S of vectors in V is a subspace of V, all you need to do is show the following are true:

• If u and v are arbitrary vectors in S, then their sum u + v is also in S.
• If u is an arbitrary vector in S and k is an arbitrary scalar in whatever field is associated with V, then ku is also in S.
• S contains the zero vector.

 

[PeroK]

Homework Statement: This is not a homework problem. I am studying abstract algebra on my own and have a question.

I know how to verify if a subset is a subspace of a vector space using the three conditions. What I am trying to imagine is a situation when one will need to verify if a subset is a subspace of a vector space.
Homework Equations: Consider a set V consist of vectors representing houses in a town, where every house is represented by a column vector with three elements :
v = ( price in $K, sq. foot, # of bathrooms )

I am assuming the set $V$ will have elements like the ones shown below.
$v_{1} = (200, 700, 2)$
$v_{2} = (250, 800, 3)$
...

1. What will be the vector space in this situation?
2. Would a subspace mean a subset of V with three or more bathrooms?

You need to go back and find some proper examples of vector spaces: polynomials make a good example.

You can't make a vector space out of housing data.

 

[WWGD]

Homework Statement: This is not a homework problem. I am studying abstract algebra on my own and have a question.

I know how to verify if a subset is a subspace of a vector space using the three conditions. What I am trying to imagine is a situation when one will need to verify if a subset is a subspace of a vector space.
Homework Equations: Consider a set V consist of vectors representing houses in a town, where every house is represented by a column vector with three elements :
v = ( price in $K, sq. foot, # of bathrooms )

I am assuming the set $V$ will have elements like the ones shown below.
$v_{1} = (200, 700, 2)$
$v_{2} = (250, 800, 3)$
...

1. What will be the vector space in this situation?
2. Would a subspace mean a subset of V with three or more bathrooms?

Most likely your data points will nit fall exactly into a line\plane , so they won't be part of a vector space , in this case Euclidean space, i assume, would be the space " hosting" the set of points you described.

 

[Stephen Tashi]

Consider a set V consist of vectors representing houses in a town, where every house is represented by a column vector with three elements :
v = ( price in $K, sq. foot, # of bathrooms )

Paying strict attention to mathematical definitions, there is a distinction between a "vector" and an n-tuple of numbers. One may often assign coordinates to a situation by representing it with an n-tuple of numbers. ( If we insist that a coordinate system define an object uniquely, your example does not give coordinates of individual houses. In you example, a 3-tuple like (200,700,2) could designate several different houses. So (200,700,2) represents all houses that share those common properties.)

One can regard a set of n-tuples of numbers as a vector space and define vector addition in the usual way - e.g. (250,800,3) + (200,700,2) = (450,1500,5). However the mathematical definitions do not define how to apply results to particular physical situations. Perhaps you can imagine some process of adding two houses together, but it's unlikely the total price would be the sum of the prices of the two houses, considering the added expense of moving the two houses to a common location. If you take (a,b,c) to represent the total value, total square footage and total bathrooms owned by a person who perhaps owns several houses then you might interpret adding two 3-tuples as giving those facts for a pair of people who combine their holdings. That would still leave the problem of interpreting calculations like 9(200,800,2) = (1800,7200,18). What would that result mean?

When people say that your example "is not a vector space" they mean that it is not obvious how the usual vector operations we define on 3-tuples could be interpreted as saying something about houses. It is true that a vector space can be defined on 3-tuples of numbers. The defect in your example is not a mathematical defect. It is a defect in the application of mathematics to a particular situation. The result of vector space operations performed on two descriptions of houses has no obvious interpretation as a description of another house..

The elementary physical intuition for a vector is that it is something that has "a magnitude and a direction". From the mathematical point of view, a vector need not have a magnitude. A large number of examples can be constructed by thinking of functions as vectors. Consider real valued functions of whose domain is $\mathbb{R}$. For functions $f$ and $g$, we have that $f+g$ is a function and $5( f + g)$ is a function. We can do manipulations like $3(f+g) = 3f + 3g$ and $(2 + 3) f = 2f + 3f$.

However if we consider functions with specialized properties, such as discontinuous functions or periodic functions, it may or may not be the case that $f$ and $g$ having the specialized property will imply that $f+g$ and $5 f$ also have it. For example, is the sum of two discontinuous functions necessarily discontinuous? Is the the sum of two periodic functions a periodic function? Such questions amount to asking whether a subset of functions with a specialized property is a subspace of the vector space of functions.

For example, let $V$ be the set of polynomials (in one variable $x$ ) of degree at most 3. $V$ is a vector space over the field of the real numbers if we define the vector operations in the "natural" manner. (Multiplying two polynomials can produce a polynomial of higher degree, but that doesn't matter because multiplication of two vectors is not a defined operation in a vector space.) Let $S$ be the subset of $V$ consisting of those polynomials that have $(x-5)$ as a factor. Is $S$ a subspace of $V$?