Section 4.4 Linear dependence and independence
Linear dependence and linear independence set of vectors are defined exactly in a same way as we defined in \(\R^n\text{.}\)
Definition 4.4.1. Linear Dependence.
A set of vectors \(\{v_1,v_2,\ldots,
v_k\}\subset V\) is said to be linearly dependent if there exists scalars \(\alpha_1,\alpha_2,\ldots \alpha_k\) not all zero such that \(\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_k v_k=0\text{.}\)
It is easy to see that \(S=\{v_1,v_2,\ldots,
v_k\}\subset V\) is linearly dependent if there exists \(i=\{1,\ldots,k\}\) such that \(v_i\in L(S\setminus \{v_i\})\text{.}\)
Definition 4.4.2. Linear Dedependence.
A set of vectors \(S=\{v_1,v_2,\ldots,
v_k\}\subset V\) is said to be linearly independent if it is not linearly dependent.
Let us understand this notion in order to get a working definition. Let us write the linearly dependent definition using quantifiers.
A set \(S=\{v_1,v_2,\ldots, v_k\}\subset V\) is linearly dependent if
\begin{equation*}
\exists (\alpha_1,\ldots,\alpha_k)\in \R^n\setminus \{0\} (\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_k v_k=0)\text{.}
\end{equation*}
\(S\) is linearly independent is same as negating the above statement. Thus
A set \(S=\{v_1,v_2,\ldots, v_k\}\subset V\) is linearly independent if
\begin{equation*}
\forall(\alpha_1,\ldots,\alpha_k)\in \R^n\setminus \{0\}(\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_k v_k\neq 0)\text{.}
\end{equation*}
The contra positive of the above statement state that A set \(S=\{v_1,v_2,\ldots,
v_k\}\subset V\) is linearly independent whenever \(\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_k v_k=0\) implies \(\alpha_1=\cdots=\alpha_k=0\text{.}\)
Thus we have the following equivalent definition of a linearly independent set.
Definition 4.4.3.
\(S=\{v_1,v_2,\ldots,
v_k\}\subset V\) is linearly independent whenever \(\alpha_1 v_1+\alpha_2 v_2+\cdots+\alpha_k v_k= 0\) implies \(\alpha_1=\cdots=\alpha_k=0\text{.}\)
Checkpoint 4.4.4.
If \(0\in S\text{,}\) then \(S\) is linearly dependent.
Checkpoint 4.4.5.
(i) If \(0\neq v\in V\text{,}\) then \(\{v\}\) is linearly independent.
(iv) \(\{u,v\}\subset V\) is linearly dependent if one of them is scalar multiple of the other.
Checkpoint 4.4.6.
(i) Let \(V={\cal P}_n(\R)\text{.}\) The set \(\{1,x,x^2,\ldots, x^n\}\) is linearly independent.
(ii) Check if \(\{1+x+x^2+x^3,x+x^3,x^2+x^3,x^3\}\) is linearly independent in \({\cal P}_3(\R)\text{.}\)
Checkpoint 4.4.7.
Let \(A\in M_n({\R})\) such that \(A^k=0\) and \(A^{k-1}\neq 0\text{.}\) Then show that
\begin{equation*}
\{I, A, A^2,\ldots, A^{k-1}\}
\end{equation*}
is linearly independent in \(M_n(\R)\text{.}\)
Checkpoint 4.4.8.
Let \(S=\{v_1,v_2,\ldots, v_k\}\subset V\) is linearly independent set. Suppose \(v\in V\) such that
\begin{equation*}
v=\alpha_1 v_1+\cdots \alpha_k v_k=\beta_1 v_1+\cdots \beta_k v_k
\end{equation*}
for scalars, \(\alpha_i\)’s and \(\beta_j\)’s. Then \(\alpha_1=\beta_1,\cdots=\alpha_k=\beta_k\text{.}\) In other words, every vector in \(V\) can be written in a unique way as a linear combination of the elements from \(S\text{.}\)
Checkpoint 4.4.9.
Let \(u,v,w\) be three vectors in \(V\text{.}\) Show that \(\{u,v,w\}\) is linearly independent if and only if \(\{u+v,u+w,v+w\}\) is linearly independent.
