Section 3.4 Linear Independence
Definition 3.13.
A set of vectors \(S\) is linearly independent if the only linear combination of elements of \(S\) that equals the zero vector is the trivial linear combination. In other words, \(S\) being a linear independent set implies that if \(c_1\vec{v_1}+c_2\vec{v_2}+...+c_k \vec{v_k}=\vec{0}\) where \(\vec{v_i} \in S\text{,}\) then all \(c_i=0\text{.}\)
A set of vectors \(S\) is linearly dependent if the set is not linearly independent. More specifically, there exists a solution to \(c_1\vec{v_1}+c_2\vec{v_2}+...+c_k \vec{v_k}=\vec{0}\) where \(\vec{v_i} \in S\) and at least one of the \(c_j \neq
0\text{.}\)
Question 3.33.
Is the set \(\left\{ \colvec{1\\ -3\\
2} \right\}\) linearly independent?
Question 3.34.
Is the set \(\left\{\colvec{2\\ 3\\ 0},
\colvec{-1\\ -1\\ 2} \right\}\) linearly independent?
Question 3.35.
(a)
Choose a vector \(\vec{v}\) so that the set \(\left\{ \colvec{2\\
3\\ 0}, \colvec{-1\\ -1\\ 2} , \vec{v} \right\} \) is linearly independent, where \(\vec{v} \in \mathbb{R}^3\text{.}\)
(b)
Is your choice of \(\vec{v}\) in \(Span \left( \left\{
\colvec{2\\ 3\\ 0}, \colvec{-1\\ -1\\ 2} \right\} \right)\text{?}\) Show why or why not.
Question 3.36.
Is \(\{ 2+t^2, 1+t^2 \}\) a linearly dependent set in \(\mathbb{P}_2\text{?}\)
Exercise 3.14.
Is \(\left\{ \begin{bmatrix} 1\amp 1\\0\amp
0 \end{bmatrix},\begin{bmatrix}0\amp 0\\ 1\amp 1
\end{bmatrix},\begin{bmatrix} 1\amp 0\\0\amp 1
\end{bmatrix},\begin{bmatrix} 0\amp 1\\1\amp 0 \end{bmatrix}
\right\}\) a linearly independent set in \(M_{2 \times 2}\text{?}\)
Exercise 3.15.
Prove that \(\{ 1+t,t+t^2,1+t^2 \}\) is linearly independent.
Theorem 3.16.
If a set \(S\) of a vector space \(V\) contains \(\vec{0}_V\text{,}\) then \(S\) is linearly dependent.
Question 3.37.
If \(A\) is a \(m\) by \(n\) matrix, then the columns of A form a linearly independent set if and only if \(A\) has pivot columns. Completely justify your response.
Theorem 3.17.
If \(M=\{ \vec{v_1},\vec{v_2},...,\vec{v_n}\}\) is linearly independent, then any subset of \(M\) is linearly independent.
Question 3.38.
Prove or disprove: If \(M=\{ \vec{v_1},\vec{v_2},...,\vec{v_n}\}\) is linearly dependent, then any subset of \(M\) is linearly dependent.
Theorem 3.18.
If \(\vec{u}\) is in the span of \(S\text{,}\) then \(S \cup \{\vec{u}\}\) is linearly dependent.
The following two theorems are a wonderful summary of the difference between and the importance of linear dependence and linear independence.
Theorem 3.19.
If \(S\) is a linearly dependent set, then any \(\vec{w} \in span(S)\) can be written as a linear combination from \(S\) in more than one way.
Theorem 3.20.
If \(S\) is a linearly independent set, then any \(\vec{w} \in span(S)\) can be written as a linear combination from \(S\) in only one way.
