DLA Terminology and notation

Section 15.3 Terminology and notation

vector addition: a rule for associating to a pair of objects $v$ and $w$ a third object $v + w$
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scalar multiplication: a rule for associating to a number $k$ and an object $v$ another object $k v$
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vector space: a collection of mathematical objects, along with appropriate conceptions of vector addition and scalar multiplication, that satisfies the Vector space axioms
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vector: an object in a vector space
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zero vector: the special vector $0$ in a vector space that satisfies vector addition Axiom A 4
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negative vector (of a vector $v$ ): the special vector $- v$ that satisfies vector addition Axiom A 5 relative to $v$
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vector subtraction: for vectors $v$ and $w,$ write $v - w$ to mean $v + (- w)$
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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
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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 11–14
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$M_{m \times n} (R)$: the vector space of all $m \times n$ matrices with entries that are real numbers; when $m = n$ we sometimes just write $M_{n} (R)$ to mean the vector space of all square $n \times n$ matrices
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$P (R)$: the vector space of all polynomials with real coefficients in a single variable
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$P_{n} (R)$: the vector space of all polynomials with real coefficients in a single variable that have degree $n$ or less
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$F (D)$: the vector space of all real-valued functions that are defined on the domain $D$
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