Motivation

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If the reader is familiar with analytic geometry, she will probably know that points in the plane can be identified by ordered tuples   where each entry is a number denoting the distance of the point from the origin in a certain direction. We call   and   the coordinates of the point in the plane, and they are often real numbers.

Although these ordered tuples are useful for describing the plane, it would seem that they lack some of the desirable behaviour of real numbers. Consider the equation  ; we know that it defines a unique number   , and we can find that number by noting that the equation is equivalent to   . Can we do the same for ordered tuples? Given an equation like  , can we identify   ? Does the equation even make sense?

As you may have guessed, this equation does make sense, and yes, we can solve it, but first we must make clear what it means to add two ordered tuples, and what it means to multiply them by a number.

Definitions

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A vector space   over a field   is a set of objects under two binary operations


 


typically called vector addition and scalar multiplication respectively, which satisfies axioms below. Note that we call elements of   vectors and elements of   scalars.

  • 1) There is a vector   such that for any   we have   (existence of additive identity).
  • 2) For any   we have   (commutativity of vector addition).
  • 3) For any   we have   (associativity of vector addition).
  • 4) For any   there is a vector   such that   where   is the additive identity mentioned above (existence of additive inverse).
  • 5) There is a scalar   such that for any   we have   (existence of multiplicative identity).
  • 6) For any vectors   and scalar   we have   (distributivity of scalar multiplication over vector addition).
  • 7) For any vector   and scalars   we have   (distributivity of scalar multiplication over field addition). Note that   is addition in the field.
  • 8) For any vector   and scalars   we have   (compatibility of field multiplication with scalar multiplication). Note that   is multiplication in the field.

Exercise: Compare these axioms with those for other algebraic structures like groups, rings, and fields. Note the similarities and differences. In what sense is a vector space like a group? How is a vector space like a field? Can you think of a structure that is both a vector space and a field?

Exercise: Although not stated explicitly, these axioms imply that   for any vector   where   is the additive identity in the field. Prove this. Hint: use axioms 1 and 7.

Examples

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  • Ordered tuples like those mentioned in the beginning of this lesson, with appropriate definitions of vector addition and scalar multiplication, can be made into elements of a vector space. Let   and define:
  
  

where   (but any field would work). This gives us definitions of vector addition and scalar multiplication in terms of field addition and field multiplication, concepts with which we are quite comfortable. Checking these definitions against the axioms should convince you that the set of ordered tuples under these operations is a vector space over  .

Now consider these definitions:

  
  

Remarkably, this defines a vector space over   as well! Although this vector space and the one previously defined share the same set of vectors, and are defined over the same field, they are different as vector spaces.

  • Any field is a vector space over itself. To see this, take the elements of the field to be the vectors, then check this against the axioms.
  • The complex numbers are a vector space over the real numbers.

Subspaces

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Any non-empty subset of is called a subspace of V if it respect the two following properties:


 

 


And we can easily verify that such a set is a vector space over  .

Linear Combinations

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Dependence Relations and Linear Independence

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Any set   of elements of   is said linearly independent (or "free") if it is every linear combination of those vectors is different from the zero vector. In other words, that :

 


or, in a completely equivalent manner :

 


Otherwise, such a set is said "dependent".

Linear Span

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Basis and Dimension

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Spanning Sets

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Vector Spaces over Finite Fields

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