Abstract vector spaces

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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, x²} 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 e₁ = 1, e₂ = x, e₃= x² be the three basis vectors. Then we could find a vector lying in the e₁ - e₂ subspace by forming some linear combination of e₁ and e₂, a vector in the e₁ - e₃ subspace by forming some linear combination of e₁ and e₁, and a vector in the e₂ -e₃ subspace by forming some linear combination of e₂ and e₃ 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, x²} as a basis for the space. Every vector in the space can be expressed as some linear combination of these three basis vectors. If e₁= 1, e₂ = x, e₃ = x² are the three basis vectors, then any vector v in the vector space can be computed as

v = a₁e₁ + a₂e₂ + a₃e₃

The n-tuple (a₁, a₂, a₃) is the coordinate vector of v with respect to the basis B. The coordinate vector of a vector v is given as [V]_B = ( a₁, a₂, a₃). 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 x². 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.