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Common Notation and Symbols in Linear Algebra Paul Skoufranis June 25, 2015 The following is an incomplete list of mathematical notation and symbols that may be used MATH 304. Shorthand Notation: ∀ for all ∃ there exists ∴ therefore s.t. such that =⇒ implies ⇐⇒ if and only if Set Notation: {objects | restrictions}. If X and Y are sets, x∈X x is an element of the set X x∈ /X x is not an element of the set X X ∪Y the union of X and Y ; that is {x | x ∈ X or x ∈ Y } X ∩Y the intersection of X and Y ; that is {x | x ∈ X and x ∈ Y } X \Y X setminus the set Y ; that is {x | x ∈ X and x ∈ / Y} ∅ the empty set (the set with no elements) X⊆Y X is a subset of Y ; that is, if x ∈ X then x ∈ Y f : X → Y a function f with domain X and codomainY Mathematical Symbols: N the natural numbers Z the integers Q the rational numbers R the real numbers C the complex numbers F an arbitrary field th (x ) Pnm n≥1 a sequence starting at n = 1 whose n term is xn a the sum a + a + · · · + a + a k k+1 m−1 m n=k n C(X) the set of continuous functions on a set X (e.g. C([0, 1]) is the set of continuous function on [0,1]) Common [ai,j ] δi,j Fn In , 0 n IV , 0V L(V, W) Mn,m (F) ker(T ) Im(T ) span(S) β [T ]γ Linear Algebra Notation: a matrix whose (i, j)th entry is ai,j the Kronecker delta function; that is δi,j = 0 if i 6= j and δi,j = 1 if i = j the set of n-tuples with entries in a field F the n × n identity and zero matrices the identity and zero operators on a vector space V the space of linear maps from V to W the set of n × m matrices with entries in F the kernel of a linear map T the range of a linear map T the span of a set of vectors S the matrix of a linear map T : V → W with respect to the basis β of V and the basis γ of W