Discover Linear Algebra: A first course in linear algebra

a linearly independent spanning set

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

the unique set of scalars $c_1,c_2,\dots ,c_n$ so that $\mathbf{w}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\cdots +c_n{\mathbf{v}}_n$

the vector $(c_1,c_2,\dots ,c_n)$ in ${\mathbb{R}}^n$ formed by the coordinates of $\mathbf{w}$ relative to $\mathcal{B}$

notation to mean the coordinate vector $(c_1,c_2,\dots ,c_n)$ in ${\mathbb{R}}^n$ for the vector $\mathbf{w}\text{,}$ relative to the basis $\mathcal{B}$ for the vector space that contains $\mathbf{w}$

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