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Abstract vector spaces

The study of abstract vector spaces is the domain of the subject "Linear Algebra". Matrix Theory deals with matrices and the vector spaces of n-dimensional Euclidean space. Although Linear Algebra employs the ideas and results of Matrix Theory extensively it is a distinctly separate subject employing a lot of radically different ideas and methods.

Let us consider abstract vector spaces. These are spaces in which the vectors are general abstract objects as opposed to just the usual n-vectors of Euclidean n-space. Any such abstract vector space of n dimensions has a basis consisting of n linearly independent vectors. Every vector in such a space can be expressed as some linear combination of its basis vectors.

Let us take an example. Consider the abstract vector space consisting of polynomials of degree
2 or less. The set {1, x, x^{2}} is a set of three linearly independent vectors that comprise a basis for
this vector space. Every vector in the space can be expressed as some linear combination of
these three basis vectors. There are an infinite number of other bases for the space which can be
formed by forming linear combinations of these three basis vectors, just as is the case for basis
vectors of n-space. As an example, let us find another set of three linearly independent vectors
that could serve as a basis. Let e_{1} = 1, e_{2 } = x, e_{3 }= x^{2} be the three basis vectors. Then we could
find a vector lying in the e_{1} - e_{2} subspace by forming some linear combination of e_{1} and e_{2}, a
vector in the e_{1} - e_{3} subspace by forming some linear combination of e_{1} and e_{1}, and a vector in
the e_{2} -_{ }e_{3} subspace by forming some linear combination of e_{2} and e_{3} and these three vectors
would be linearly independent and constitute a basis.

There is a difference between the n-vectors of Matrix Theory and those of an abstract linear
space. In Matrix Theory the vectors that we deal with are specified directly as the n-tuple that
they are. When we refer to the n-tuple, we refer to the vector itself. In Linear Algebra the
abstract vectors we deal with are not generally specified directly, but rather indirectly, through
their "coordinate vector". The actual vector can be computed if desired but we generally specify
them by their coordinate vector as referred to a specified basis. Let us take the example we used
above. Consider the abstract vector space consisting of polynomials of degree 2 or less. Let us
take the set B = {1, x, x^{2}} as a basis for the space. Every vector in the space can be expressed as
some linear combination of these three basis vectors. If e_{1}= 1, e_{2} = x, e_{3} = x^{2} are the three basis
vectors, then any vector v in the vector space can be computed as

v = a_{1}e_{1} + a_{2}e_{2} + a_{3}e_{3}

The n-tuple (a_{1}, a_{2}, a_{3}) is the coordinate vector of v with respect to the basis B. The coordinate
vector of a vector v is given as [V]_{B} = ( a_{1}, a_{2}, a_{3}). So we refer to vectors by these coordinate
vectors. Suppose the coordinate vector of v is [V]_{B} = (2, 3, 5). Then the actual vector v is 2 +
3x + 5 x^{2}. We can compute the vector out to get its actual value if we wish but it is usually
sufficient to just use its coordinate vector.

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