Discover Linear Algebra

Section 17.2 Terminology and notation

subspace

a subset of vectors in a vector space that itself is a vector space under the same addition and scalar multiplication operations as the parent vector space

trivial subspace

the subspace of a vector space consisting of only the zero vector

linear combination (of a collection of vectors $\uvec{v}_1,\uvec{v}_2,\dotsc,\uvec{v}_m$)

a vector that can be expressed as

$$\begin{equation*} k_1\uvec{v}_1 + k_2\uvec{v}_2 + \dotsb + k_m\uvec{v}_m \end{equation*}$$

for some collection of scalars $k_1,k_2,\dotsc,k_m$

subspace generated by a set of vectors $S$

the subspace of a vector space consisting of all possible linear combinations of vectors in $S\text{;}$ also called the span of $S$, and written $\Span S$

spanning set (for a vector space)

a set of vectors in a vector space (or subspace of a vector space) where the subspace generated by the set is in fact the whole space; could also be called a generating set of vectors for the space

solution space of homogeneous system $A\uvec{x}=0$

the subspace of $\R^n$ (where $n$ is the number of columns of $A$) consisting of all solutions to the system