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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 \uvec{w} relative to a basis \basisfont{B}=\{\uvec{v}_1,\uvec{v}_2,\dotsc,\uvec{v}_n\}

the unique set of scalars c_1,c_2,\dotsc,c_n so that \uvec{w}=c_1\uvec{v}_1+c_2\uvec{v}_2+\dotsb+c_n\uvec{v}_n

coordinate vector associated to a vector \uvec{w} relative to a basis \basisfont{B}

the vector (c_1,c_2,\dotsc,c_n) in \R^n formed by the coordinates of \uvec{w} relative to \basisfont{B}

\rmatrixOf{\uvec{w}}{B}

notation to mean the coordinate vector (c_1,c_2,\dotsc,c_n) in \R^n for the vector \uvec{w}\text{,} relative to the basis \basisfont{B} for the vector space that contains \uvec{w}

\matrixOf{\uvec{w}}{B}

notation to mean the coordinate vector in \R^n for the vector \uvec{w}\text{,} relative to the basis \basisfont{B} for the vector space that contains \uvec{w}\text{,} realized as a column vector (i.e. as a column matrix)