Section 2.5 Subspace of a vector space
We define subspace of a vector space in this section.
Remark 2.5.2.
A subspace \(W\) of \(V\) is written as \(W\leq V\text{,}\) and if it is a proper subspace (i.e., \(W\leq V\) and \(W\neq V\)) then it is written as \(W< V\text{.}\)We give following criterion to check whether a given nonempty subset of a vector space is a subspace or not.
Theorem 2.5.3.
Let \(V\) be a vector space over a field \(F\text{.}\) A nonempty subset \(W\) of \(V\) is subspace if and only if \(W\) is closed under addition and scalar multiplication or equivalently
\begin{equation*}
\text{if }\alpha,\beta\in F \text{ and } v,w\in W\text{ then, }\alpha v+\beta w\in W.
\end{equation*}
Example 2.5.4. (Trivial subspace).
Example 2.5.5. (A subspace of \(n\)-dimensional vector space).
\begin{equation*}
W=\big\{\alpha_1 e_{i_1}+\alpha_2 e_{i_2}+\cdots\alpha_r e_{i_r}:\alpha_i\in F\text{ and } i_j\in\{1,2,\ldots,n\} \big\}.
\end{equation*}
This is a subspace of \(F^n\text{.}\)Example 2.5.6. (A subspace of \(n\times n\) matrices).
\begin{equation*}
D_n(F)=\big\{{\rm diag}(a_1,a_2,\ldots,a_n):a_i\in F\big\}
\end{equation*}
is a subspace of \(M_{n\times n}(F)\text{.}\)Example 2.5.7. (A subspace of the direct sum of two vector spaces).
\begin{equation*}
S=\big\{ (v,0):v\in V\big\}
\end{equation*}
is a subspace of \(V\bigoplus W\text{.}\)Example 2.5.8. (Intersection of subspaces).
\begin{equation*}
\bigcap_{i\in I}S_i
\end{equation*}
is a subspace of \(V\text{.}\)Example 2.5.9. (Sum of subspaces).
\begin{equation*}
\sum_{i\in I}S_i=\bigg\{v_1+v_2+\cdots+v_r:v_j\in\bigcup_{i\in I}S_i \text{ and }r\in\N \bigg\}
\end{equation*}
is a subspace.Example 2.5.10. (Solutions of homogeneous system of linear equations).
\begin{equation*}
\ker(A)=\big\{X\in M_{n\times 1}(F): AX=0 \big\}
\end{equation*}
is a subspace of \(M_{n\times 1}(F)\text{.}\)Definition 2.5.11. (Subspace spanned by a nonempty subset).
Let \(S\) be a nonempty subset of a vector space \(V\text{.}\) The subspace spanned by \(S\) is the intersection of all subspaces containing \(S\text{.}\) We denote it by \(\langle S\rangle\) or by \(\Span(S)\text{.}\)By Example 2.5.8, the intersection of subspaces is again a subspace. Note that a subspace spanned by \(S\) is the smallest subspace of \(V\) containing \(S\text{.}\)
