We are given a vector where and are real numbers. We would like to write as a linear combination of vectors and This means we are looking to find three real numbers and such that

(1)

The right side of the above equality tells us

(2)

This means

(3)

Therefore

(4)

This seems like a very trivial example but it constitutes the main result in our future studies.

Suppose we are given the vector and we would like to write this vector as a linear combination of two vectors and . We are looking for two values and such that

(5)

The right side of the above equality tells us

(6)

Therefore

(7)

which means

(8)

Of course we know from previous lessons how to solve this system of linear equations in many different ways to get

(9)

Therefore

(10)

Suppose we are given the set of all polynomials of the form where and are real numbers. Take the usual addition operation for polynomials and multiplication of a polynomial with a constant, together with the given set. We show the set and the two operation constitute a vector space.

a. Suppose we have two polynomials and . Then

(11)

which is a polynomial in

b.

(12)

c. Given another polynomial we have

(13)

d. Take the zero polynomial and observe that

(14)

Note that the zero polynomial is equal to zero for all values of .

e. Take the polynomial and observe that

(15)

f. Suppose is a real number. Then

(16)

which is a polynomial in

g. Observe that

(17)

h. Given a real number

(18)

i. Observe that

(19)

j. Last thing

(20)

We checked all ten properties of a vector space.

The set of integers and the two operations addition and multiplication do not construct a vector space since given the real number and the element of integers, we have

(21)

which does not belong to the set of integers.

We want to determine if the set of all vectors such that is a subspace of . Note that in order to show a nonempty set is a subspace of a vector space , we need to show is closed with respect to both operations inherited from First note that the set provided by this example is nonempty since belongs there. Now suppose and are two members of this set and is a real number. Since we can write

(22)

where . Therefore the result belongs to the given set. Next,

(23)

where This shows the given line is a subspace of

We would like to check if the set spans Using the definition of an spanning set, we need to check if every vector in can be expressed as a linear combonation of the elements of the given set. Let’s pick an arbitrary vector and write

(24)

This means

(25)

that is

(26)

Therefore

(27)

Therefore any vector in can be written as a linear combination

(28)

which means this set spans