What is a vector space?
A vector space v is a set of vectors
with two operations defined,
addition and scalar multiplication,
which satisfy the axioms of addition
and scalar multiplication.
And there are five axioms of addition
and five axioms of scalar multiplication
that must be satisfied in order for a set of vectors
to be a vector space.
Let's first take a look at the axioms of addition.
If we let vector v, vector w and vector z
be vectors in a vector space,
then they must satisfy the following axioms of addition.
Number one closed under addition.
If vector v and vector w are in the vector space v ,
then vector v plus vector w is also in the vector space.
The commutative law of addition.
vector v + vector w must equal vector w+vector v .
The associative law of addition.
The sum of vector v and w + vector v
must equal vector v + the sum of vector w and vector z.
The existence of additive identity,
which in, in a KT0 vector must be in the vector space,
such that vector v + a zero vector equals vector v.
And then we have the existence of an additive inverse,
which means each vector in these set of vectors
must have the opposite vector
such as vector v + the opposite of vector v
is equal to the zero vector.
And now let's take a look
at the five axioms of salar multiplication.
Where now a and b are real numbers.
And again, vector v vector w in vector z,
are vectors in the vector space.
First, we have closed under scalar multiplication,
which means if a is a real number
and v is in the vector space,
then a times vector v is also in the vector space.
Next a times the sum of vector v and w
is equal to a times vector v + a times vector w.
Next we have the sum of a and b times vector v
is equal to a times vector v + b times vector v.
Next, we have a times the product of b in vector v
equals the product of a and b times vector v.
And then finally we have 1 times vector v equals vector v.
So again, all of 10 of these axioms
must be satisfied in order for a set of vectors,
to be a vector space.
I included this slide just to show all 10 axioms
in one list.
Now let's take a look at some examples of vector spaces,
again, to prove these are vector spaces.
We would have to go through all 10 axioms and
show they are true.
First, Rn is a vector space
where Rn is a collection of ordered list
or n-tupples X sub one through X sub n.
Where X sub j is a real number for j =1 through n.
So they st up vectors using all possible n-tupples
makes up our n which is a vector space.
The set of all 2 by 3 matrices is a vector space.
The set of polynomials of degree n is a vector space,
in the form shown below as Pn
and the set of functions that map
real numbers to real numbers,
is also a vector space.
And again, to show all these are vector spaces,
we would have to go through all 10 axioms
and show they hold true.
Before we go I do want to mention
the definition does define the two
operations vector addition denoted by the + sign
and scalars multiplication denoted by
placing a scalar next to the vector.
However, vector space,
did not use usual operations.
And for this reason,
the operations will always be given
in the definition of the vector space.
Just keep in mind,
the 10 axioms must still hold true
under the definitions of the operations,
in order for these set of vectors
to be a vector space.
I hope you found this helpful.