Section 16.3 Terminology and notation
- vector addition
a rule for associating to a pair of objects \(\uvec{v}\) and \(\uvec{w}\) a third object \(\uvec{v}+\uvec{w}\)
- scalar multiplication
a rule for associating to a number \(k\) and an object \(\uvec{v}\) another object \(k\uvec{v}\)
- vector space
a collection of mathematical objects, along with appropriate conceptions of vector addition and scalar multiplication, that satisfies the Vector space axioms
- vector
an object in a vector space
- zero vector
the special vector \(\zerovec\) in a vector space that satisfies vector addition Axiom A 4
- negative vector (of a vector \(\uvec{v}\))
the special vector \(-\uvec{v}\) that satisfies vector addition Axiom A 5 relative to \(\uvec{v}\)
- vector subtraction
for vectors \(\uvec{v}\) and \(\uvec{w}\text{,}\) write \(\uvec{v}-\uvec{w}\) to mean \(\uvec{v}+(-\uvec{w})\)
- trivial vector space
a vector space that consists of a single object, which then must be the zero vector in that space; also called the zero vector space
Here follows the notation we will use for some common vector space examples.
- \(\R^n\)
the usual vector space of \(n\)-tuples of real numbers that we have been studying in Chapters 12–15
- \(\matrixring_{m \times n}(\R)\)
the vector space of all \(m\times n\) matrices with entries that are real numbers; when \(m=n\) we sometimes just write \(\matrixring_n(\R)\) to mean the vector space of all square \(n\times n\) matrices
- \(\poly(\R)\)
the vector space of all polynomials with real coefficients in a single variable
- \(\poly_n(\R)\)
the vector space of all polynomials with real coefficients in a single variable that have degree \(n\) or less
- \(F(D)\)
the vector space of all real-valued functions that are defined on the domain \(D\)