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What is the difference between a Set, a Vector, and a Vector Space?

Please, explain without using formal math notations. Because if you use formal math notations, I wouldn't understand anything.

Domates
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3 Answers3

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A set is what's called a primitive notion. That means we don't actually define it, we just assume that everyone has roughly the same idea in mind when we say it. Think what a set of objects means outside of math. Same definition should work here. A set is a group(/ collection/ assortment/ assemblage/ ... gaggle -- maybe that one only works for geese) of objects. Those objects are called members or elements of the set.

A vector is a member of a vector space.

A vector space is a set of objects which can be multiplied by regular numbers and added together via some rules called the vector space axioms.

An example of a set is $\{1,2,3\}$. This is just a collection of three numbers. Another example is $\{banana, orange, taco, oatmeal\}$. This set contains $4$ foods. Neither one of these sets comes pre-equipped with any notion of addition or multiplication -- they're just groups of objects.

An example of a vector space is the set of all real numbers, $\Bbb R$, along with the usual kind of multiplication and addition. This is a set of numbers, but we also can talk about adding them and multiplying them by other numbers. You can check that they satisfy all of those rules that adding and multiplying have to satisfy and thus this set is also a vector space.
The number $3$ is an example of a vector in the above vector space. So is $\pi$. So is $\dots$

A less trivial (and more typical) example of a vector space is the set of all continuous functions defined on the interval $[0,1] \subset \Bbb R$. Addition of two functions $f$ and $g$ is defined by $(f+g)(x) = f(x)+g(x)$ and multiplication is defined by $(af)(x) = a\cdot f(x)$, for $a\in \Bbb R$. You can verify that this is also a vector space.
The function $f$ given by $f(x)=x^2-2$ is a vector is this vector space.

  • If vector-space is a set, doesn't it already incorporate those axioms from the set theory? –  Oct 02 '15 at 20:53
  • If u update the answer with this information and add some examples, I would accept your answer. Coz, it is almost complete. –  Oct 02 '15 at 20:58
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A vector space is a set in which you can add its elements with one another and in which you can "stretch" its elements, in other words multiply them by a real number. The elements of the vector space are the vectors. Some people would say vector spaces are sets with "more structure".

On the other hand vector spaces are different from most sets in the sense that their elements might be arbitrary and it might no be able to add them with each other.

Asinomás
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A set is a collection of objects. For example, the set of integers from $1$ through $5$.

A vector space is a set of elements (called vectors) which is defined "over a field" in the sense that if you multiply by a number in the field (think real numbers), you still get an element in the vector space. It is also possible to add things together in a vector space, and there is a special element called '$0$' which has the property that $\vec{v}+0=\vec{v}=0+\vec{v}$ for every element (same thing as a vector) $\vec{v}$ in the vector space.

Ben Sheller
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