Discover Linear Algebra

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

$\R^n$

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 $D$