Difference between revisions of "Vector space"
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| − | A '''vector space''' is a | + | A '''vector space''' is a construct in mathematics that generalizes the familiar notion of vectors in the x-y plane. Recall that given two vectors <math>\vec{v} = (v_1,v_2)</math> and <math>\vec{w} = (w_1,w_2)</math>, we can take form the sum <math>\vec{v}+\vec{w}=(v_1+v_2,w_1+w_2)</math> of two vectors, and the product of a vector with a scalar (i.e., a real number), by setting <math> a \cdot \vec{v} = (av_1,av_2)</math>. A vector space consists of a collection of objects that has two analogous operations: it is possible to add two of the objects together, and it is possible to multiply one by a scalar. Vector spaces are the fundamental objects of study of [[linear algebra]]. |
| − | The | + | == Examples == |
| + | # The space <math>\mathbb R^n</math> of n-tuples of real numbers is a vector space, where to add two vectors we simply add the corresponding components. The case <math>n=2</math> is exactly the case of vectors in the plane discussed above. | ||
| + | # The set <math>\mathbb R[x]</math> of polynomials with real coefficients is a vector space. If we add two polynomials together, we get another polynomial, and similarly, if we multiply a polynomial by a constant, we get another polynomial. Note that although it's also possible to multiply two polynomials and get another one, this is not part of the vector space structure: a vector space with a reasonable notion of multiplication of vectors is called an [[algebra]]. | ||
| + | # The set of polynomials of degree less than or equal to <math>n</math> (for any <math>n \geq 0</math>) is a vector space, for the same reason. | ||
| + | # The set of all continuous functions on the real line is a vector space: the sum of two continuous functions is again continuous, as is the product of a continuous function with a constant. | ||
| + | # The set of <math>2 \times 2</math> matrices <math>M_{2 \times 2}(\mathbb R)</math> is a vector space. | ||
| + | # If <math>V</math> and <math>W</math> are vector spaces, then we can form a new vector space <math>V \oplus W</math> (called the direct sum of V and W) whose entries are ordered pairs <math>(v,w)</math> of elements of V and elements of W. | ||
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| + | == Properties == | ||
| + | Many familiar properties of vectors in the plane carry over to the set of vector spaces. For example, just as the plane is 2-dimensional, it makes sense to talk about the dimension of any vector space (though it may be infinite, as in the case of polynomials!) Vectors in the plane can all be written in the form <math>ae_1+be_2</math>, where <math>e_1 = (1,0), e_2=(0,1)</math>, and a set elements with this same property that all vectors can be written as sums of multiples of vectors in the set is called a basis. Having a convenient basis often makes computations easier. It turns out that every finite dimensional vector space has a basis -- in fact, if we're feeling adventurous and assume the Axiom of Choice, even every infinite dimensional vector space has a basis. | ||
| + | |||
| + | However, a general vector space has no notion of "distance": given a vector, there's not necessarily a way to define the length <math>|v|</math> of that vector. For example, it's not obvious how we should define the length of a polynomial or a matrix. A vector space endowed which a notion of distance is called ''normed''. | ||
| + | |||
| + | == Generalizations == | ||
| + | The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars. More generally, given any field <math>k</math>, a vector space over <math>k</math> is an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold. In terms of another definition, a vector space is simply a module for which the ground ring is a field. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 21:07, October 31, 2009
A vector space is a construct in mathematics that generalizes the familiar notion of vectors in the x-y plane. Recall that given two vectors 
and 
, we can take form the sum 
of two vectors, and the product of a vector with a scalar (i.e., a real number), by setting 
. A vector space consists of a collection of objects that has two analogous operations: it is possible to add two of the objects together, and it is possible to multiply one by a scalar. Vector spaces are the fundamental objects of study of linear algebra.
Examples
- The space
of n-tuples of real numbers is a vector space, where to add two vectors we simply add the corresponding components. The case 
is exactly the case of vectors in the plane discussed above. - The set
of polynomials with real coefficients is a vector space. If we add two polynomials together, we get another polynomial, and similarly, if we multiply a polynomial by a constant, we get another polynomial. Note that although it's also possible to multiply two polynomials and get another one, this is not part of the vector space structure: a vector space with a reasonable notion of multiplication of vectors is called an algebra. - The set of polynomials of degree less than or equal to
(for any 
) is a vector space, for the same reason. - The set of all continuous functions on the real line is a vector space: the sum of two continuous functions is again continuous, as is the product of a continuous function with a constant.
- The set of
matrices 
is a vector space. - If
and 
are vector spaces, then we can form a new vector space 
(called the direct sum of V and W) whose entries are ordered pairs 
of elements of V and elements of W.
Properties
Many familiar properties of vectors in the plane carry over to the set of vector spaces. For example, just as the plane is 2-dimensional, it makes sense to talk about the dimension of any vector space (though it may be infinite, as in the case of polynomials!) Vectors in the plane can all be written in the form 
, where 
, and a set elements with this same property that all vectors can be written as sums of multiples of vectors in the set is called a basis. Having a convenient basis often makes computations easier. It turns out that every finite dimensional vector space has a basis -- in fact, if we're feeling adventurous and assume the Axiom of Choice, even every infinite dimensional vector space has a basis.
However, a general vector space has no notion of "distance": given a vector, there's not necessarily a way to define the length 
of that vector. For example, it's not obvious how we should define the length of a polynomial or a matrix. A vector space endowed which a notion of distance is called normed.
Generalizations
The above discussion only considers vector spaces in which the scalars are real numbers, but we could just as well talk about the set of polynomials with complex coefficients, where we multiple by complex scalars. More generally, given any field 
, a vector space over is an additive group in which addition is commutative and with which is associated a field of scalars, as the field of real numbers, such that the product of a scalar and an element of the group or a vector is defined, the product of two scalars times a vector is associative, one times a vector is the vector, and two distributive laws hold. In terms of another definition, a vector space is simply a module for which the ground ring is a field.
