Vector Spaces

Definition

  • Vector Space: A vector space, denoted as V, is a set equipped with two operations: vector addition and scalar multiplication, satisfying specific properties.

Properties of Vector Spaces

  1. Closure under Addition: For any vectors u,vV their sum u+v is also in V.
  2. Associativity of Addition: Addition is associative; (u+v)+w=u+(v+w) for all u,v,wV.
  3. Commutativity of Addition: u+v=v+ufor all u,vV.
  4. Existence of Additive Identity: There exists a vector 0 in V such that v+0=v for all vV.
  5. Existence of Additive Inverse: For every vV, there exists a vector v in V such that v+(v)=0.
  6. Closure under Scalar Multiplication: For any scalar c and vector v in V, the scalar multiplication cv is in V.
  7. Distributivity of Scalar over Vector Addition: c(u+v)=cu+cv for all scalars c and vectors u,vV.
  8. Distributivity of Scalars: (c+d)v=cv+dv for all scalars c,d and vV.
  9. Compatibility of Scalar Multiplication with Field Multiplication: c(dv)=(cd)v for all scalars c,d and vV.
  10. Scalar Multiplication Identity: 1v=v for all vV, where 1 is the multiplicative identity of the field.

Examples of Vector Spaces

  1. Euclidean Space: The set of n-tuples of real numbers Rn equipped with standard addition and scalar multiplication.
  2. Polynomial Space: The set of all polynomials of degree at most n with coefficients from a field.
  3. Function Space: The set of all real-valued functions on a given interval [a,b] with operations defined pointwise.

Subspaces

  • Subspace: A subset W of a vector space V 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 Transformations

  • Linear Transformation: A function T:VW between two vector spaces that preserves vector addition and scalar multiplication. It satisfies T(u+v)=T(u)+T(v) and T(cv)=cT(v) for all u,vV and scalars c.