Objectives: Topics
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Linear independence
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Geometric interpretation of linearly independent vectors
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Section 1.7 Linear Independence
Worksheet 1.7 Linear Independence
Objectives: Goals
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Characterize a set of vectors and linear systems using the concept of linear independence.
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Construct dependence relations between linearly dependent vectors.
Motivating Question: What is the smallest number of vectors needed in a parametric solution to a linear system?
Definition 1.44.
A set of vectors \({\vec v_1, \ldots, \vec v_k}\) in \(\mathbb{R}^n\) is called linearly independent if the vector equation
\begin{equation*}
c_1 \vec v_1 + c_2 \vec v_2 + \cdots + c_k \vec v_k = \vec 0
\end{equation*}
has only the trivial solution \(c_1 = c_2 = \cdots = c_k = 0\text{.}\) The set \({\vec v_1, \ldots, \vec v_k}\) is said to be linearly dependent if there exist weights \(c_1, c_2, \dots, c_k\text{,}\) not all zero, such that
\begin{equation*}
c_1 \vec v_1 + c_2 \vec v_2 + \cdots + c_k \vec v_k = \vec 0
\end{equation*}
Let \(A = \begin{bmatrix} \vec v_1 \amp \cdots \amp \vec v_n \end{bmatrix}\text{.}\) Then some equivalent definitions of linear independence are
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\(A\vec x = \vec 0\) only has the trivial solution.
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\(A\) has a pivot in every column.
Example 1.45.
For each of the following sets, determine if the set is linearly independent or linearly dependent.
(a)
\(\left\{\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix} 2 \\ 2 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} \right\}\)(b)
\(\left\{\begin{bmatrix}1 \\ 2 \\ 1\end{bmatrix}, \begin{bmatrix} 2 \\ 4 \\ 2 \end{bmatrix}\right\}\)
(c)
\(\left\{\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix} 2 \\ 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \right\}\)
