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Section 16.2 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
