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)