Skip to main content

Section 1.4 Spanning and Linear Independence

Definition 1.4.1.

A vector \(\vect{v}\) is a linear combination of the vectors: \(\vect{v_1}\text{,}\) \(\vect{v_2}\text{,}\) … \(\vect{v_n}\) if \(\vect{v} = t_1\vect{v_1}+t_2\vect{v_2} + \cdots + t_n\vect{v_n}\) for some choice of real numbers \(t_i\text{.}\) The \(t_i\) are called coefficients.

Definition 1.4.2.

The set of all linear combinations of the vectors from non-empty set \(\mathbb{S} = \{\vect{v_1}, \vect{v_2}, \ldots, \vect{v_n} \}\) is called the span of \(\mathbb{S}\text{,}\) and written:

\begin{equation*} span\mathbb{S} = \{ t_1\vect{v_1}+t_2\vect{v_2}+\cdots+t_n\vect{v_n} | \; t_i \in \mathbb{R} \} \end{equation*}

Is \(\ot{1}{0}{-1}\) in the span of \(\mathbb{S}=\{\ot{1}{2}{3}, \ot{4}{5}{6}, \ot{7}{8}{9} \}\text{?}\) If so write \(\ot{1}{0}{-1}\) as a linear combination of the vectors in \(\mathbb{S}\text{.}\)

For each of the following sets \(\mathbb{S}\text{,}\) give a geometric description of \(span\mathbb{S}\text{.}\)

  1. \(\displaystyle \mathbb{S}=\{ \ot{0}{1}{0} \}\)

  2. \(\displaystyle \mathbb{S}=\{\ot{5}{-2}{7} \}\)

  3. \(\displaystyle \mathbb{S}=\{\ot{0}{0}{0} \}\)

  4. \(\displaystyle \mathbb{S}=\{\ot{1}{0}{0},\ot{0}{1}{1}\}\)

  5. \(\displaystyle \mathbb{S}=\{\ot{6}{3}{-9},\ot{-4}{-2}{6}\}\)

  6. \(\displaystyle \mathbb{S} =\{\ot{1}{0}{0},\ot{0}{1}{0},\ot{0}{0}{1}\}\)

  7. \(\displaystyle \mathbb{S} =\{\ot{1}{0}{0},\ot{1}{1}{0},\ot{1}{1}{1}\}\)

  8. \(\displaystyle \mathbb{S} =\{\ot{2}{0}{2},\ot{0}{3}{0},\ot{1}{1}{1}\}\)

When new vectors are made using linear combinations of vectors from a given set, \(\mathbb{S}\text{,}\) an important questions is: “Can all vectors in \(\mathbb{R}^3\) be expressed as a linear combination of the vectors in \(\mathbb{S}\text{?}\)”

Definition 1.4.5.

A set of vectors in \(\mathbb{R}^3\) is said to span \(\mathbb{R}^3\) if \(span\mathbb{S}=\mathbb{R}^3\text{.}\) This means that every vector in \(\mathbb{R}^3\) is a linear combination of the vectors in \(\mathbb{S}\text{.}\) (Span has now been defined as a noun and as a verb.)

Determine algebraically if the following sets span \(\mathbb{R}^3\text{.}\) Showing a set spans \(\mathbb{R}^3\) requires showing that any point \(\vect{a} \in \mathbb{R}^3\) can be written as a linear combination of the vectors in the given set. Specifically, for \(\mathbb{S} = \{\vect{v_1},\vect{v_2},\ldots,\vect{v_n}\}\text{,}\) let \(\vect{a}=\ot{a}{b}{c}\in\mathbb{R}^3\) and find values for the real numbers \(t_1,t_2,\ldots,t_n\) such that \(\vect{a}=t_1\vect{v_1}+t_2\vect{v_2}+\cdots+t_n\vect{v_n}\text{.}\) If it is not the case that any point in \(\mathbb{R}^3\) can be written as a linear combination of the vectors in \(\mathbb{S}\text{,}\) describe the set of points that can. Where possible write \(\vect{a}=\ot{1}{2}{3}\) as a linear combination of the vectors in \(\mathbb{S}\) in more than one way.

  1. \(\displaystyle \mathbb{S} =\{\ot{1}{2}{0},\ot{0}{0}{1},\ot{2}{4}{6} \}\)

  2. \(\displaystyle \mathbb{S} =\{\ot{1}{0}{0},\ot{1}{1}{0},\ot{1}{1}{1} \}\)

  3. \(\displaystyle \mathbb{S} =\{\ot{1}{0}{0},\ot{1}{1}{0},\ot{1}{1}{1},\ot{0}{1}{1}\}\)

  4. \(\displaystyle \mathbb{S} =\{\ot{1}{1}{0},\ot{1}{1}{1}\}\)

Definition 1.4.7.

A set \(\mathbb{S} = \{\vect{v_1}, \vect{v_2}, \ldots, \vect{v_n} \}\) is linearly independent if for any point in \(span\mathbb{S}\) the choice of coefficients used to describe that point as a linear combination of the elements of \(\mathbb{S}\) is unique.

Definition 1.4.8.

A set that both spans \(\mathbb{R}^3\) and is linearly independent is called a basis for \(\mathbb{R}^3\text{.}\)

The most familiar basis is \(\beta=\{\ot{1}{0}{0},\ot{0}{1}{0},\ot{0}{0}{1}\}\) where \(\ot{1}{2}{3}=1\ot{1}{0}{0}+2\ot{0}{1}{0}+3\ot{0}{0}{1}\text{.}\) However sometimes it may be useful to use \(\alpha=\{\ot{1}{1}{1},\ot{0}{1}{1},\ot{0}{0}{1}\}\text{,}\) making \(\ot{1}{2}{3}=1\ot{1}{1}{1}+1\ot{0}{1}{1}+1\ot{0}{0}{1}\text{.}\) This is called making a change of basis and the list of coefficients written:

\begin{equation*} \begin{array}{ccc} [\ot{1 }{2}{3}]_{\beta} =\left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right] \amp \amp [\ot{1 }{2}{3}]_{\alpha} =\left[ \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right] \end{array} \end{equation*}