A basis of a vector space is a set of vectors in that space that can be used as coordinates for it. The two conditions such a set must satisfy in order to be considered a basis are
- the set must span the vector space;
- the set must be linearly independent.
A set that satisfies these two conditions has the property that each vector may be expressed as a finite sum of multiples of basis elements in precisely one way (up to reordering). The "multiples" of the basis elements may be thought of as the coordinates of the vector, with respect to that basis.
Bases (the plural of basis) are used to translate the language of linear algebra into the language of matrices. They are solely responsible for the connection between linear transformations and matrices, the usual interpretation of linear transformations. The properties of bases provide the framework for a variety of important properties of vector spaces and linear transformations, like dimension and rank.
The so-called elementary vectors demonstrate the connection between coordinates and bases. In, for instance, four-dimensional Euclidean space the vector space the vectors form a basis for . There is only one way to write any element of as a sum of real multiples of those four elements; for instance,
The span of a set of vectors seeks to describe the set of all possible vectors that could be reached by performing the usual vector space operations on vectors in . It turns out that this "span" is a vector space itself.
A set of vectors (non-uniquely) defines a subspace of by creating all possible linear combinations of the elements of . So,
Main Article: Linear Independence
The notion of linear independence encapsulates the notion of "redundancy" in the creation of a set's span. If a set of vectors could span the same space with one less element, then it is known as linearly dependent; if it could not span the same space with one less element, then it is known as linearly independent. Equivalently, the zero-vector may be expressed as a linear combination of elements of in precisely one way: with coefficients of zero.
The vector space of subtractive color mixing is created from real multiples of vectors , , and with the understanding that . (When mixing colors, red, yellow, and blue combine to make black.)
For instance, some elements of the vector space are However, since they differ by a multiple of . They differ by
What is true of the following sets of vectors in this vector space?
Let be a set that spans and is linearly independent. Then, any vector in may be expressed as a linear combination of elements of in only one way.
By definition of span, any vector in may be expressed as a linear combination of elements of .
By definition of linear independence, may be expressed as a linear combination of elements of in only one way. Now suppose a vector can be expressed as a linear combination of elements of in two different ways: But then and not all terms cancel out on the righthand side (otherwise, the linear combinations would be the same). This contradicts the definition of linear independence.
It follows that any vector in may be expressed as a linear combination of elements of in only one way.
This theorem reconciles the definition of a basis with its crucial property. It is also necessary to show that there do, in fact, exist bases for arbitrary vector spaces, but that follows from mathematical induction for finite-dimensional vector spaces and Zorn's lemma for infinite-dimensional vector spaces. The properties of linearity provide a strong groundwork for further results, like those regarding the "size" of a vector space.
In order to discuss the "dimension" of a vector space, it is important to realize that this is not a defining concept of vector space. While many examples like have a very immediate concept of dimension (the number of coordinates, ), there are a myriad of others where it is not so clear. And indeed, it is even surprising that certain vectors form a basis!
The real vector space of "Fibonacci-like" sequences contains all sequences satisfying for all natural numbers . Two sequences can be added by adding their corresponding terms, and a sequence can be multiplied by a scalar by multiplying all elements in the sequence by a real number. (Note that the resulting sequence in each case still satisfies the Fibonacci property and is therefore still in the vector space.)
The Fibonacci numbers and the Lucas number are both elements of this sequence. The two geometric series and , where and are also elements of this sequence.
What makes a basis for this vector space?
Suppose is a basis for the vector space , and suppose is another basis for . Then and have the same cardinality.
If is finite, then the dimension of is known as the number of elements in ,
If there exists a basis with a finite number of elements, then an injective linear transformation from the vector space to itself must be a bijection. This can be shown by mathematical induction (on the number of elements in the finite basis) and is necessary for the following proof.
Suppose is finite and smaller than . Then contains vectors . Consider the linear transformation defined by for each integer , . The idea is to show that this mapping is injective but not surjective, creating a contradiction (with the above fact).
Let be a vector such that . Then, for some scalars , so Since the 's are linearly independent, it follows that for all and therefore . Then, must be injective, since its kernel is .
Now, suppose is a vector such that . There must be such a vector in order for to be surjective. Then, for some scalars , so Since the 's are linearly independent, there is no possible choice of 's that satisfy this equation. Then, must not be surjective.
However, in a vector space with a finite basis, an injection must be surjective. It follows that cannot be finite if is larger than it. In other words, and must have the same number of elements. This holds for infinite bases too, but the argument requires more knowledge of infinite sets. (In particular, a -union of finite sets has cardinality .)
Main Article: Change of Basis