AATA Definitions and Examples

Section 20.1 Definitions and Examples

A vector space

V

over a field

F

is an abelian group with a scalar product

α \cdot v

α v

defined for all

α \in F

and all

v \in V

satisfying the following axioms.

$α (β v) = (α β) v;$
$(α + β) v = α v + β v;$
$α (u + v) = α u + α v;$
$1 v = v;$

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where

α, β \in F

and

u, v \in V .

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The elements of

V

are called vectors; the elements of

F

are called scalars. It is important to notice that in most cases two vectors cannot be multiplied. In general, it is only possible to multiply a vector with a scalar. To differentiate between the scalar zero and the vector zero, we will write them as 0 and

0,

respectively.

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Let us examine several examples of vector spaces. Some of them will be quite familiar; others will seem less so.

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Example 20.1.

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The

n

-tuples of real numbers, denoted by

R^{n},

form a vector space over

R .

Given vectors

u = (u_{1}, \dots, u_{n})

and

v = (v_{1}, \dots, v_{n})

R^{n}

and

α

R,

we can define vector addition by

u + v = (u_{1}, \dots, u_{n}) + (v_{1}, \dots, v_{n}) = (u_{1} + v_{1}, \dots, u_{n} + v_{n})

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and scalar multiplication by

α u = α (u_{1}, \dots, u_{n}) = (α u_{1}, \dots, α u_{n}) .

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Example 20.2.

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F

is a field, then

F [x]

is a vector space over

F .

The vectors in

F [x]

are simply polynomials, and vector addition is just polynomial addition. If

α \in F

and

p (x) \in F [x],

then scalar multiplication is defined by

α p (x) .

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Example 20.3.

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The set of all continuous real-valued functions on a closed interval

[a, b]

is a vector space over

R .

f (x)

and

g (x)

are continuous on

[a, b],

then

(f + g) (x)

is defined to be

f (x) + g (x) .

Scalar multiplication is defined by

(α f) (x) = α f (x)

for

α \in R .

For example, if

f (x) = \sin x

and

g (x) = x^{2},

then

(2 f + 5 g) (x) = 2 \sin x + 5 x^{2} .

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Example 20.4.

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Let

V = Q (\sqrt{2}) = {a + b \sqrt{2} : a, b \in Q} .

Then

V

is a vector space over

Q .

u = a + b \sqrt{2}

and

v = c + d \sqrt{2},

then

u + v = (a + c) + (b + d) \sqrt{2}

is again in

V .

Also, for

α \in Q,

α v

is in

V .

We will leave it as an exercise to verify that all of the vector space axioms hold for

V .

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Proposition 20.5.

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Let

V

be a vector space over

F .

Then each of the following statements is true.

$0 v = 0$ for all $v \in V .$
$α 0 = 0$ for all $α \in F .$
If $α v = 0,$ then either $α = 0$ or $v = 0 .$
$(- 1) v = - v$ for all $v \in V .$
$- (α v) = (- α) v = α (- v)$ for all $α \in F$ and all $v \in V .$

Proof.

To prove (1), observe that

0 v = (0 + 0) v = 0 v + 0 v;

consequently,

0 + 0 v = 0 v + 0 v .

Since

V

is an abelian group,

0 = 0 v .

The proof of (2) is almost identical to the proof of (1). For (3), we are done if

α = 0 .

Suppose that

α \neq 0 .

Multiplying both sides of

α v = 0

1 / α,

we have

v = 0 .

To show (4), observe that

v + (- 1) v = 1 v + (- 1) v = (1 - 1) v = 0 v = 0,

and so

- v = (- 1) v .

We will leave the proof of (5) as an exercise.

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