6.4.1 Span of 3 Vectors Example

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Span of 3 Vectors

Let's consider these three vectors.

\begin{aligned} {\vec{a}}_{1} = [\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}] & {\vec{a}}_{2} = [\begin{array}{c} 0 \\ 0 \\ - 1 \\ - 1 \end{array}] & {\vec{a}}_{3} = [\begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}] \end{aligned}

Analysis. If $\vec{b} \in Span {{\vec{a}}_{k}}_{k = 1}^{3}$ , then

\begin{aligned} \vec{b} & = \sum_{k = 1}^{3} x_{k} \cdot {\vec{a}}_{k} \\ = x_{1} [\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}] + x_{2} [\begin{array}{c} 0 \\ 0 \\ - 1 \\ - 1 \end{array}] + x_{3} [\begin{array}{c} 1 \\ 1 \\ 1 \\ 1 \end{array}] \\ = [\begin{array}{c} x_{1} \\ x_{1} \\ 0 \\ 0 \end{array}] + [\begin{array}{c} 0 \\ 0 \\ - x_{2} \\ - x_{2} \end{array}] + [\begin{array}{c} x_{3} \\ x_{3} \\ x_{3} \\ x_{3} \end{array}] \\ = [\begin{array}{c} x_{1} + x_{3} \\ x_{1} + x_{3} \\ - x_{2} + x_{3} \\ - x_{2} + x_{3} \end{array}] \end{aligned}

How do we know if a vector is within the span? Let's rewrite the span with Greek letters to give us a clearer picture. Let

\vec{b} = [\begin{matrix} α \\ α \\ β \\ β \end{matrix}]

for some $α, β \in R$ . Consider:

[\begin{matrix} - 1 \\ - 1 \\ 2 \\ 3 \end{matrix}] \notin Span {{\vec{a}}_{k}}_{k = 1}^{3}

since the vector is not in the form

\vec{b} = [\begin{matrix} α \\ α \\ β \\ β \end{matrix}]

However, these vectors are in the span:

[\begin{matrix} 1 \\ 1 \\ - 3 \\ - 3 \end{matrix}], [\begin{matrix} 24 \\ 24 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}]

This third vector is a special case. It was vector ${\vec{a}}_{3}$ in the span of $\vec{b}$ . However, ${\vec{a}}_{3}$ is redundant with respect to the linear combination.

${\vec{a}}_{3}$ adds no new information to the span.
${\vec{a}}_{3}$ is linearly dependent on ${{\vec{a}}_{1}, {\vec{a}}_{2}}$ .
${\vec{a}}_{3}$ is a linear combination of ${\vec{a}}_{1}$ and ${\vec{a}}_{2}$ .