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Section 2.7 Linear Independence

Definition 2.7.1.

A subset \(\mathbb{S} = \{ \mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_n\}\) of a linear space \(\mathbb{L}\) is linearly independent if every point \(\mathbf{Q}\) in \(span\mathbb{S}\) can be written in only one way as a linear combination using elements of \(\mathbb{S}\text{.}\) More specifically, for any real numbers \(a_1,a_2,\ldots,a_n\) and \(b_1,b_2,\ldots,b_n\text{:}\)

\begin{equation*} a_1\mathbf{P}_1 + a_2\mathbf{P}_2 +\cdots+a_n\mathbf{P}_n = b_1\mathbf{P}_1 + b_2\mathbf{P}_2 +\cdots+ b_n\mathbf{P}_n \Longrightarrow a_1=b_1,\ldots a_n=b_n \end{equation*}

A subset \(\mathbb{S}\) is linearly dependent if \(\mathbb{S}\) is not linearly independent.

Show how to write the point \(\ot{a}{b}{c}\) as a linear combination of the points in \(\mathbb{S} = \{ \ot{1}{2}{0},\ot{1}{0}{0},\ot{1}{1}{0} \}\text{.}\) Is the set \(\mathbb{S}\) linearly independent? If it is not, show how some point can be written in more than one way as a linear combination of the elements of \(\mathbb{S}\text{.}\)

What is the additive identity element in the linear space \(\mathbb{R}^n\text{?}\) ...in the linear space \(\mathbb{P}\text{?}\) ..in the linear space C[0,1]?

Describe the steps needed to prove:

\begin{equation*} (A \Longrightarrow B) \Longleftrightarrow (C \Longrightarrow D) \end{equation*}