2.3	Span of a Set of Vectors

Kristen Pueschel

2.3 Span of a Set of Vectors

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Definition: The span of a set of vectors

{{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{k}}

is the set of all linear combinations

{c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{k} {\vec{v}}_{k} | c_{1}, \dots, c_{k} \in R}

. In other words, the span is all of the vectors that you can build by scaling and summing the vectors

{{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{k}}

.
We begin with a few examples of checking if a vector is in the span of a set of vectors.

Finally we look at ways to describe the span of a set of vectors:

Proposition

The set

{{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{k}}

of vectors in

R^{n}

will span

R^{n}

only if the reduced row echelon form of the matrix

A = [{\vec{v}}_{1} {\vec{v}}_{2} \dots {\vec{v}}_{k}]

has a pivot in each row.

Consider:

Is the vector $\vec{v} = [\begin{matrix} π \\ e \end{matrix}]$ in the span of { $[\begin{matrix} 1 \\ 2 \end{matrix}]$ , $[\begin{matrix} 2 \\ 1 \end{matrix}]$ }? Describe this span.

Is the vector $\vec{v} = [\begin{matrix} 2 \\ 3 \\ 5 \end{matrix}]$ in the span of { $[\begin{matrix} 4 \\ 6 \\ 10 \end{matrix}]$ , $[\begin{matrix} 1 \\ 0 \\ 2 \end{matrix}]$ }? Describe this span.

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