Vector Spaces
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Vectors, dimension and bases
The number of vectors in any basis for a given vector space is constant and is equal to the dimension of the vector space. In fact, we define the dimension of a vector space to be the number of vectors in the basis. Of course, this only applies if the vector space has a finite spanning set; that is, if the vector space is finite dimensional. If it is finite dimensional then the number of vectors in any basis for V is its dimension. However; we would need to prove that any basis for a vector space V has the same number of vectors. Two examples of 2-dimensional vector spaces and their standard bases include [Equation goes here - download the original to see it.]A typical element is[Equation goes here - download the original to see it.] The standard basis is[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] A typical element is[Equation goes here - download the original to see it.] The standard basis is[Equation goes here - download the original to see it.] A question that arises is: are these two vector spaces really examples of one and the same abstract structure? In order to answer this question we need some way of comparing the two vector spaces. This process of comparison is provided by the concept of a vector space isomorphism. This is a mapping (a function) that preserves the vector space structure. That means the mapping must take the result of adding vectors in one vector space to the result of adding vectors in one vector space to the result of adding vectors in the other vector space; it must also preserve the result of multiplying vectors by scalars. Formally, a vector space isomorphism is a one-one mapping (a bijection) between vector spaces V and W such that, for any two vectors[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] A vector space isomorphism creates a one-one correspondence between the two vector spaces. This means that if two vector spaces are isomorphic then a linearly independent set of vectors in one space corresponds to a linearly independent set of vectors in the other. Hence, an isomorphism maps a basis for one vector space onto a basis for the other. So if two finite dimensional vector spaces are isomorphic they must have the same dimension. The converse also applies - if two finite vector spaces have the same dimension they are isomorphic. An isomorphism between finite vector spaces V and W can be specified as a mapping between elements of a basis for V and elements of a basis for W. For example, [Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.]can be shown to be isomorphic by the correspondence [Equation goes here - download the original to see it.] An isomorphism between[Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.]is [Equation goes here - download the original to see it.]
The number of vectors in any basis for a given vector space is constant and is equal to the dimension of the vector space. In fact, we define the dimension of a vector space to be the number of vectors in the basis. Of course, this only applies if the vector space has a finite spanning set; that is, if the vector space is finite dimensional. If it is finite dimensional then the number of vectors in any basis for V is its dimension. However; we would need to prove that any basis for a vector space V has the same number of vectors. Two examples of 2-dimensional vector spaces and their standard bases include [Equation goes here - download the original to see it.]A typical element is[Equation goes here - download the original to see it.] The standard basis is[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] A typical element is[Equation goes here - download the original to see it.] The standard basis is[Equation goes here - download the original to see it.] A question that arises is: are these two vector spaces really examples of one and the same abstract structure? In order to answer this question we need some way of comparing the two vector spaces. This process of comparison is provided by the concept of a vector space isomorphism. This is a mapping (a function) that preserves the vector space structure. That means the mapping must take the result of adding vectors in one vector space to the result of adding vectors in one vector space to the result of adding vectors in the other vector space; it must also preserve the result of multiplying vectors by scalars. Formally, a vector space isomorphism is a one-one mapping (a bijection) between vector spaces V and W such that, for any two vectors[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] A vector space isomorphism creates a one-one correspondence between the two vector spaces. This means that if two vector spaces are isomorphic then a linearly independent set of vectors in one space corresponds to a linearly independent set of vectors in the other. Hence, an isomorphism maps a basis for one vector space onto a basis for the other. So if two finite dimensional vector spaces are isomorphic they must have the same dimension. The converse also applies - if two finite vector spaces have the same dimension they are isomorphic. An isomorphism between finite vector spaces V and W can be specified as a mapping between elements of a basis for V and elements of a basis for W. For example, [Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.]can be shown to be isomorphic by the correspondence [Equation goes here - download the original to see it.] An isomorphism between[Equation goes here - download the original to see it.]and[Equation goes here - download the original to see it.]is [Equation goes here - download the original to see it.]
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Contents of
Vector Spaces
1 Vector Spaces
2 Linear polynomials
3 Complex numbers
4 Quadratic polynomials
5 Continuous Functions
6 Infinite sequences
7 Vector space axioms
8 Properties of vector spaces
9 Subspaces of vector spaces
10 Basis and dimension
11 Span
12 Theorem (Vector spaces)
13 Vectors, dimension and bases
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