The vector space is a set of vectors
with addition and scalar multiplication
that satisfy the axioms
of addition and scalar multiplication shown below.
We have five axioms of addition
and five axioms of scalar multiplication.
In this example, as it like all the following
that are vector spaces.
First we have the set of real numbers
with the standard operations
of addition and scalar multiplication.
To see if this set is a vector space,
we need to verify all five axioms of addition
and all five axioms of scalar multiplication.
If we take a look at these axioms though,
if we replace vectors u and v with real numbers,
they should remind us of several of the properties
of real numbers.
For example,
the sum of two real numbers is equal to another real number.
And therefore we have Closure under Addition,
the Commutative Law of Addition
and the Associative Law of Addition.
all hold true for real numbers.
Zero was a real number
and therefore the Additive Identity is true
where a real number plus zero equals to real number.
And every little number has its opposite,
and therefore the existence
of an Additive Inverse is also true.
And then for the axioms of scalar multiplication,
for Closure under Scalar Multiplication,
if a is a real number,
A times, another real number is still a real number,
both Distributive Properties hold true for real numbers.
As well as the Associative Property,
and one is a real number where one times, a real number
is equal to the same real number,
which satisfies the Multiplicative Identity property.
The set of real numbers is the vector space known as our one
or just our.
Next we have the set of all three by four matrices
with the standard operations of addition
and scalar multiplication.
Let's take a closer look at this set.
If we add matrix A and matrix B,
which are both two by three matrices,
we get another two by three matrix,
which we can call matrix C,
by adding the corresponding elements.
And therefore we do have Closure under Addition.
Matrix addition is also Commutative and Associative.
There is a zero two by three matrix,
such a matrix A plus zero matrix is equal to matrix A,
which verifies the existence of the Additive Identity.
And for every matrix A,
there is the opposite of matrix A
where the elements have the opposite sign
such that the sum of the two,
two by three matrices
is equal to the two by three zero matrix.
Moving onto the axioms of scalar multiplication.
If c is a scalar,
c times matrix A is equal to another two by three matrix
which we can call a matrix B
where the elements of matrix B are found
by multiplying c and every element in matrix A.
We have closure under Scalar Multiplication.
The Distributive properties are also true,
for all two by three matrices,
where c times the sum of matrix A and B is equal to c times
matrix A plus c times matrix B,
and the quantity c plus d times matrix A,
is equal to c times matrix A plus d times matrix A.
Similarly,
the Associative Property multiplication also holds true.
And one times matrix A is equal to matrix A,
verifying the Multiplicative Identity property.
The set of two by three matrices makes up a vector space.
Next, we have the set of all integers,
with the standard operations of addition
and scalar multiplication.
So looking at the axioms of addition,
if we add two integers we still get an integer.
We have Closure under Addition.
Integer addition is Commutative and Associative that's true.
Zero is an integer,
and therefore we do have,
the existence of the Additive Identity.
Every integer has its opposite,
which verifies we have the existence of an Additive Inverse.
But moving on to the Axioms of Scalar Multiplication,
we run into a problem.
Remember A and B are any real number.
So if we take any real number and multiply it by an integer,
we do not always get another integer.
And therefore we don't have Closure
under Scalar Multiplication.
For example, for example,
if we let A equal one half,
and multiply by the integer, let's say three,
this gives us three halves or 1.5,
which is not in the set of integers,
which is why we don't have Closure
under Scalar Multiplication we can stop.
The set of integers is not a vector space.
Next we have the set of all polynomials of degree,
exactly four,
again with standard operations of addition,
and scalar multiplication.
So, all of the polynomials in this set
or in the form of P of X,
equals A sub zero, plus A sub one times X,
plus A sub two times X squared,
plus A sub three times X cubed,
plus A sub four times X to the power fourth.
However, if all the polynomials have degree exactly four,
we also have the condition that A sub four,
can't equal zero.
Because if it did, the degree will be less than four.
Starting with the axioms of addition and all looks good
until we get to the existence of the Additive Identity.
The zero polynomial would be P of X equals zero.
But P of X equals zero has degree zero.
And therefore the zero polynomial
is not in the set of all polynomials of degree exactly four,
and therefore we can stop.
We don't have the existence of an Additive Identity.
And therefore the set of polynomials,
of degree exactly four is not a vector space.
And then finally we have the set all Continuous Functions
defined on the close interval from one to three,
again with the standard operations of addition
and scalar multiplication.
Let's take a closer look.
We first checked for Closure under Addition.
So if we add two Continuous Functions,
over the close interval from one to three,
we would get a new function,
which we can call H of X,
which will still be Continuous,
over the closed interval from one to three,
we do have Closure under Addition.
Remember, when adding functions,
we're simply adding the Y values,
which would not affect whether the function was continuous
or not.
Function Addition is also Commutative and Associative,
for the existence of the Additive Identity,
the function Y equals zero is Continuous,
over the closed interval from one to three
And therefore the Additive Identity does exist in the set,
where if f[x] plus zero equals f[x],
and for every continuous function
over the closed interval from one to three,
there is the opposite of the function
such that the sum is equal to zero,
which satisfies the existence of an Additive Inverse.
And now we move on to the axioms of scalar multiplication.
Any scalar a times,
a Continuous Function,
over the closed interval from one to three,
results in another function,
continuous over the closed interval from one to three.
we have Closure under Scalar Multiplication.
The Distributive Properties are also true,
Where c times the sum of f[x] and g[x],
equals c times f[x], plus c times g[x],
and the quantity c plus d times f[x],
is equal to c times f[x], plus d times f[x].
The Associative Property multiplication is also true,
and we also have the multiplicative identity,
where one time f[x] is equal to f[x].
All the axioms are satisfied.
The set of all Continuous Functions,
defined over the close interval from one to three,
does form a vector space.
I did brush over some of these axioms fairly quickly.
So you may want to go back
and verify some of these in more detail.
I hope you found this helpful.