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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