- Vector Space: A vector space, denoted as , is a set equipped with two operations: vector addition and scalar multiplication, satisfying specific properties.
Properties of Vector Spaces
- Closure under Addition: For any vectors their sum is also in .
- Associativity of Addition: Addition is associative; for all .
- Commutativity of Addition: for all .
- Existence of Additive Identity: There exists a vector in such that for all .
- Existence of Additive Inverse: For every , there exists a vector in such that .
- Closure under Scalar Multiplication: For any scalar and vector in , the scalar multiplication is in .
- Distributivity of Scalar over Vector Addition: for all scalars and vectors .
- Distributivity of Scalars: for all scalars and .
- Compatibility of Scalar Multiplication with Field Multiplication: for all scalars and .
- Scalar Multiplication Identity: for all , where is the multiplicative identity of the field.
Examples of Vector Spaces
- Euclidean Space: The set of -tuples of real numbers equipped with standard addition and scalar multiplication.
- Polynomial Space: The set of all polynomials of degree at most with coefficients from a field.
- Function Space: The set of all real-valued functions on a given interval with operations defined pointwise.
- Subspace: A subset of a vector space that is itself a vector space under the same operations (vector addition and scalar multiplication).
Basis and Dimension
- Basis: A set of vectors that spans a vector space and is linearly independent.
- Dimension: The number of vectors in a basis of a vector space; it represents the "size" of the vector space.
Linear Independence and Span
- Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.
- Span: The set of all possible linear combinations of a given set of vectors.
- Linear Transformation: A function between two vector spaces that preserves vector addition and scalar multiplication. It satisfies and for all and scalars .