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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 .
If
\(0\neq v\in V\text{,}\) then
\(\{v\}\) is linearly independent.
\(\{u,v\}\subset V\) is linearly dependent if one of them is scalar multiple of the other.
Checkpoint 4.4.6 .
Let
\(V={\cal P}_n(\R)\text{.}\) The set
\(\{1,x,x^2,\ldots, x^n\}\) is linearly independent.
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.
