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Section 19.2 Terminology and notation

basis for a vector space

a linearly independent spanning set

ordered basis

a basis where the basis vectors are always written in a particular order, and linear combinations of the basis vectors are always expressed in that order

coordinates of a vector w relative to a basis B={v1,v2,…,vn}

the unique set of scalars c1,c2,…,cn so that w=c1v1+c2v2+⋯+cnvn

coordinate vector associated to a vector w relative to a basis B

the vector (c1,c2,…,cn) in Rn formed by the coordinates of w relative to B

(w)B

notation to mean the coordinate vector (c1,c2,…,cn) in Rn for the vector w, relative to the basis B for the vector space that contains w

[w]B

notation to mean the coordinate vector in Rn for the vector w, relative to the basis B for the vector space that contains w, realized as a column vector (i.e. as a column matrix)