Vector Spaces | Research & Encyclopedia Articles

Vector Spaces

The primary example of a vector space is Rⁿ, the set of ordered n-tuples of real numbers. If v = (v₁,...,v_n) w =(w₁,..,w_n) are elements of Rⁿ then v + w is defined to be (v₁ + w₁,...,v_n + w_n). If r is a real number, then rv is defined to be (rv₁, rv₂,..., rv_n). The set of real numbers is the field of scalars for Rⁿ. In general, a vector space V over a field K (called the field of scalars) is a set with an operation and an 'action of K'. The operation is denoted by +. It defines vector addition. The 'action of K' defines how an element of K can be multiplied with an element of V. The following rules must also be satisfied for any elements x, y, z in V and k in K:

• 1. x + y is an element of V.
• 2. kx is an element of V, where kx denotes k "times" x.
• 3. kx = xk.
• 4. x + y = y + x.
• 5. x + (y + z) = (x + y) + z.
• 6. k(x + y) = kx + ky.
• 7. There is an element denoted by 0 in V such that 0 + v = v for all v in V.
• 8. 0x = 0.
• 9. 1x = x.

When working with vectors, it is often useful to have a basis. This is a set of vectors B₁ = {v₁, v₂,...} such that any vector w in V can be written uniquely as w = k₁v₁ + k₂v₂ + ... in which the k_i's are scalars. The uniqueness referred to means that if w also equals r₁v₁ + r₂v₂ +..., then r₁ = k₁, r₂ = v₂, and so on. If two bases for V are finite, then they have the same number of elements. This fact can be proved using Gaussian elimination (see the article of Systems of Linear Equations for details). In fact, any two bases for V have the same cardinality but this fact is harder to prove if the bases are infinite. The cardinality of any base for V is called the dimension of V. For example, one basis for Rⁿ is {e₁,..., e_n} where, for any i, e_i is the vector whose coordinates are all 0 except for the i-th coordinate which is one. So the dimension of Rⁿ is n. Infinite dimensional vector spaces are common in functional analysis. One such vector space, F, is the set of all integrable functions from [0,1] to the real numbers. Two such functions, f and g, say, can be added by adding their values at each point. So (f+g)(x) is defined to be f(x) + g(x). For any real number r, (rf)(x) is defined to be rf(x). The vector space F is also a Hilbert space. Banach spaces and Hilbert spaces are vector spaces with additional properties that are used frequently in functional analysis.

Vector space homomorphisms, also known as linear transformations, are maps between vector spaces (which are defined over the same field K). They are used to show relationships among vector spaces. Precisely, a homomorphism from a vector space V to a vector space W is a map H from V to W that satisfies the following property. For all vectors x and y in V and for all scalars r, H(x + y) = H(x) + H(y) and H(rx) = rH(x). For example, there is a natural homomorphism from R² to R³ given by H((x,y)) = (x,y,0). If B₁={v₁, v₂,...} and B₂ = {w₁, w₂,...} are bases for a vector space V, then there is a homomorphism H from V to V such that H(vi) = wi for each i. Thus if x = a1v₁ + a2v₂ + ... is any vector in V, then H(x) = a1w₁ + a2w₂ + ... In this case, H can be referred to as the change-of-basis transformation. If V is n-dimensional, then H can be represented as an n x n matrix with nonzero determinant. In fact, any n x n matrix M with nonzero determinant is a change of basis matrix since the columns of M form a basis for V.

Every vector space V has a dual vector space that is usually denoted by V*. V* is the set of all homomorphisms from V to the field K. If f and g are in V*, then (f + g)(v) is defined to be f(v) + g(v). If k is in K, then (kf)(v) is defined to be kf(v). For example, in R³, if v is any vector, then the map that sends every x in R³ to v·x is in R³*(see the article on Vector Algebra for the definition of v·x). In fact, every element of R³* is of this form. The map that sends each element f of F to the integral of f over [0,1] is an element of F*.