A linear space, also called a vector space, is a set with well-defined properties. First, a linear space must be defined over a field. This field is often the field of real numbers or the field of complex numbers, but may be other fields as well. The field elements are known as scalars, and the elements of the linear space are known as vectors. (These vectors should not be confused with physical vectors of two- or three-dimensions--although those vectors are elements of linear spaces.) This space needs to have defined at least two operations, addition and scalar multiplication, that fit the following criteria:
Addition must be commutative. This means that for all pairs x,y in a linear space L, x + y = y + x.
Addition must also be associative. For all x,y,z in L, (x + y) + z = x + (y + z).
Zero exists. There must be some element within the vector space so that x + 0 = x. Under these conditions, most randomly chosen groups of numbers are not linear spaces, as many do not contain zero.
There is an opposite element to each element. That is, for every x, there must be a y such that x + y = 0.
There must be an element for unitary multiplication. When using a vector space, one wants to have 1x = x.
Multiplication must hold in three ways. First, for any scalar a,b and any vector x, (ab)x = a(bx). Then, it must also be true that (a + b)x = ax + bx. Finally, for a scalar a and any vectors x and y, it must hold that a(x + y) = ax + ay.
Most of the rules that define a linear space seem self-evident and somewhat redundant to people who are familiar with the workings of real numbers, complex numbers, or even two- or three-dimensional physical vectors. Matrices and polynomials also follow the rules of linear spaces. However, many spaces may be defined that do not fulfill these properties. Further, there are many familiar operations that do not have to be defined in a linear space. For one of the most obvious, elements of a linear space do not have to have multiplication between them defined.
There are many other features that may apply to linear spaces. A subspace may be defined by taking some elements from the original linear space and making sure that the above requirements are still met. The basis of a linear space is the set of linearly independent elements that can be linearly combined to form all other elements of the linear space. The dimension of a linear space is the number of elements in its basis.
Linear spaces can be used within linear algebra to determine properties of a system. An understanding of linear spaces will help with matrix manipulation and solving systems of equations, and is necessary for most higher and abstract mathematics. Further, comprehending linear spaces conceptually can be used in various quantum mechanical systems. Most quantum systems can be expressed as linear spaces, often giving insight into deep or interesting properties.
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