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Course by: Mark A. Davenport. E-mail the author

# Vector Spaces

Module by: Mark A. Davenport. E-mail the author

Metric spaces impose no requirements on the structure of the set MM. We will now consider more structured MM, beginning by generalizing the familiar concept of a vector.

## Definition 1

Let KK be a field of scalars, i.e., K=RK=R or CC. Let VV be a set of vectors equipped with two binary operations:

1. vector addition: +:V×VV+:V×VV
2. scalar multiplication: ·:K×VV·:K×VV

We say that VV is a vector space (or linear space) over KK if

• VS1: VV forms a group under addition, i.e.,
• (x+y)+z=x+(y+z)(x+y)+z=x+(y+z) (associativity)
• x+y=y+xx+y=y+x (commutativity)
• 0V0V such that xVxV, x+0=0+x=xx+0=0+x=x
• xVxV, yy such that x+y=0x+y=0
• VS2: For any α,βKα,βK and x,yVx,yV
• α(βx)=(αβ)xα(βx)=(αβ)x (compatibility)
• (α+β)(x+y)=αx+αy+βx+βy(α+β)(x+y)=αx+αy+βx+βy (distributivity)
• 1K1K such that 1x=x1x=x

## Example 1

• RNRN over RR (not RNRN over CC)
• CNCN over CC or CNCN over RR
• Set of polynomials of degree NN with rational coefficients over QQ
• The set of all infinitely-long sequences of real numbers over RR
• GF(2)N:{0,1}NGF(2)N:{0,1}N over {0,1}{0,1} with mod 2 arithmetic (Galois field)
• C[a,b]C[a,b] over RR

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