Vector Spaces

We generalize the theory of vectors in

by extracting all the essential features of vector addition and scaling that make the theory works. The following is the most fundamental definition in linear algebra, which embodies all the important characteristics of vectors. Definition: A vector space over

is a nonempty set

of objects, called vectors, on which are defined two operations, called addition and multiplication by scalar (real number), that satisfy the following axioms: For any

and

and real numbers

and

and are in (Closed under addition and scalar multiplication)
(commutativity of addition)
(associativity of addition)
There exists a zero vector in such that (additive identity)
For each in , there exists a vector in such that (additive inverse)
(distributivity for vector addition)
(distributivity for scalar addition)
(Compatibility of scalar multiplication with real number multiplication)
(Identity of scalar multiplication)

Remark: More generally, we can consider scalars other than

in the above definition e.g. a vector space over

. We can deduce the following simple facts directly from the above axioms:

Zero vector is unique.
For any vector , is unique. Moreover, .
for any vector . (Note: the left zero is a real number and the right zero is the zero vector.)
for any real number .

Examples of Vector Spaces

The following are some examples of vector spaces over

: Example 1: The first obvious example is

. Example 2: Polynomials For non-negative integer

the set of polynomials in

of degree at most

with real coefficients. Let

and

be polynomials in

. Then we define the following: Addition:

Scalar multiplication:

for any real number

We consider such polynomials as "vectors". It can easily be shown that they satisfy all the axioms in the definition. Zero polynomial acts as the zero vector. Therefore,

is a vector space. Example 3: Sequences of real numbers Let

be the set of all real number sequences. We denote a real number sequence

. For any sequences

and

, we define the following: Addition:

Scalar multiplication:

for any real number

Again, it is easy to verify that

is a vector space. Example 4: Matrices Let

be the set of all m x n matrices. And we have already defined the addition and scalar multiplication of m x n matrices before. It is clear that

is a vector space. Hence, the set of all linear transformation from

is also a vector space. Example 5: Real-valued functions Let

be the set of all real-valued functions defined on a set

. Let

and

be two such functions in

. Then we define the following: Addition:

for any

Scalar multiplication: Let

be any real number.

for any

The real-valued functions can be regarded as "vectors" and it can be shown that they satisfy all the axioms in the definition of vector spaces. Therefore,

is a vector space.

Exercise

Check the box if the set is a vector space. (Note: You can check multiple boxes.)

Select all that apply

A
The set of real numbers; the usual addition and scalar multiplication
B
The set of all polynomials with real coefficients; the usual polynomial addition and scalar multiplication
C
The set of all polynomials of degree 3, together with the zero polynomial; the usual polynomial addition and scalar multiplication
D
The set of all 2 x 2 matrices with zero determinant; the usual matrix addition and scalar multiplication
E
The set of all 2 x 2 upper triangular matrices; the usual matrix addition and scalar multiplication
F
The set consisting of a single vector ; and for all in
G
The set of all geometric sequences i.e. , where for some real numbers and ; addition and scalar multiplication in
H
The set of all arithmetic sequences i.e. , where for some real numbers and ; addition and scalar multiplication in

Check my answer (3)

Vector Spaces

Vector Spaces

Examples of Vector Spaces

Exercise

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