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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 \(D\)