6.3 Span of a set of vectors

#Type/Definition #Topic/Linear_Algebra #Type/Example

Types: N/A
Examples: N/A
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Generalizations: N/A

Properties: 6.4 Linearly Dependent Vectors, 6.7 Linearly Independent Vectors
Sufficiencies: N/A
Questions: N/A

Span of a set of vectors

Let $m, n \in N$ and ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{m}$ . The span of this set of vectors ${{\vec{a}}_{k}}_{k = 1}^{n}$ is the set of all possible linear combinations of these vectors, denoted as

Span {{\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n}} = {\sum_{k = 1}^{n} x_{k} {\vec{a}}_{k} : x_{k} \in R for k = 1, 2, \dots, n}

We denote the span of vectors ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n}$ using angle brackets:

⟨ {\vec{a}}_{1}, {\vec{a}}_{2}, \dots, a_{n} ⟩ = Span {{\vec{a}}_{k}}_{k = 1}^{n} .

Remark. Later, we will call this the $Rng (f)$ of the 10 Matrix-Vector Multiplication problem.
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Visual Interpretation of Span

Let

\vec{x} = [\begin{matrix} 2 \\ 1 \end{matrix}]

and visualize this vector as a single vertex.

Solution. The span of this vector is the set

⟨ \vec{x} ⟩ = {α \vec{x} : α \in R} = {[\begin{matrix} 2 α \\ 1 α \end{matrix}] : α \in R}

This is the line given by $y = 0.5 x$ having a slope of $0.5$ and y-intercept $(0, 0)$ . We graph this span below as the red line.

y=0.5x|red
(2,1)|label:(2,1)

Remark. Notice this corresponds with our interpretation of 4.1 Scalar-Vector Multiplication as a scaling of the vector value.
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Remark. Later, we will call this the Rng(f) of the 10 Matrix-Vector Multiplication problem. ⧫

Remark. Later, we will call this the $Rng (f)$ of the 10 Matrix-Vector Multiplication problem.
$⧫$