3.1 Definitions of vector spaces
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A7. (αβ)x = α(βx) for any scalars α and β and any x ∈ V. A8. 1 ·x = x for all x ∈ V.
Closure properties
An important component of the definition is the closure properties of the two operations. These properties can be summarized as follows: C1. If x ∈ V and α is a scalar, then αx ∈ V. C2. If x, y ∈ V, then x + y ∈ V.
The set V, together with the operations of addition and scalar multiplication, is said to form a vector space if the following axioms are satisfied:
A1. x + y = y + x for any x and y in V. A2. (x + y) + z = x + (y + z) for any x, y, and z in V. A3. There exists an element 0 in V such that x + 0 = x for each x ∈ V. A4. For each x ∈V, there exists an element −x in V such that x + (−x) = 0. A5. α(x + y) = αx + αy for each scalar α and any x and y in V. A6. (α + β)x = αx + βx for any scalars α and β and any x ∈ V.
Definitions of vector spaces
Introduction
The operations of addition and scalar multiplication are used in many diverse contexts in mathematics.
Regardless of the context, however, these operations usually obey the same set of algebraic rules.
Leabharlann Baidu
Examples of vector spaces
Euclidean vector spaces Rn，n 1.
C[a, b], the set of all real-valued functions that are defined and continuous on the closed interval [a, b].
Pn，n 0 the set of all polynomials of degree less
Determine whether a set U is a vector space
Given a set U on which the operations of addition and scalar multiplication have been defined and satisfy properties C1 and C2, then we must check to see if the eight axioms are valid in order to determine whether U is a vector space.
Outline
1. Definition of vector space 2. Examples of vector spaces 3. Fundamental properties of vector spaces
Definition of vector space
Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique element x + y that is also in V, and with each element x in V and each scalar α, we can associate a unique element αx in V. Namely, linear combination αx + βy in V.
Introduction
Vector space is not just a mathematical concept. The results about vector spaces have some applications. In this section we will give the definition of a vector space, show some examples and discuss some fundamental properties of vector spaces.
Thus, a general theory of mathematical systems involving addition and scalar multiplication will be applicable to many areas in mathematics.
Mathematical systems of this form are called vector spaces or linear spaces.
Closure properties
An important component of the definition is the closure properties of the two operations. These properties can be summarized as follows: C1. If x ∈ V and α is a scalar, then αx ∈ V. C2. If x, y ∈ V, then x + y ∈ V.
The set V, together with the operations of addition and scalar multiplication, is said to form a vector space if the following axioms are satisfied:
A1. x + y = y + x for any x and y in V. A2. (x + y) + z = x + (y + z) for any x, y, and z in V. A3. There exists an element 0 in V such that x + 0 = x for each x ∈ V. A4. For each x ∈V, there exists an element −x in V such that x + (−x) = 0. A5. α(x + y) = αx + αy for each scalar α and any x and y in V. A6. (α + β)x = αx + βx for any scalars α and β and any x ∈ V.
Definitions of vector spaces
Introduction
The operations of addition and scalar multiplication are used in many diverse contexts in mathematics.
Regardless of the context, however, these operations usually obey the same set of algebraic rules.
Leabharlann Baidu
Examples of vector spaces
Euclidean vector spaces Rn，n 1.
C[a, b], the set of all real-valued functions that are defined and continuous on the closed interval [a, b].
Pn，n 0 the set of all polynomials of degree less
Determine whether a set U is a vector space
Given a set U on which the operations of addition and scalar multiplication have been defined and satisfy properties C1 and C2, then we must check to see if the eight axioms are valid in order to determine whether U is a vector space.
Outline
1. Definition of vector space 2. Examples of vector spaces 3. Fundamental properties of vector spaces
Definition of vector space
Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique element x + y that is also in V, and with each element x in V and each scalar α, we can associate a unique element αx in V. Namely, linear combination αx + βy in V.
Introduction
Vector space is not just a mathematical concept. The results about vector spaces have some applications. In this section we will give the definition of a vector space, show some examples and discuss some fundamental properties of vector spaces.
Thus, a general theory of mathematical systems involving addition and scalar multiplication will be applicable to many areas in mathematics.
Mathematical systems of this form are called vector spaces or linear spaces.