span

The span of a set of vectors $\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{\\mathbf{v_{n}}}$ of a vector space $V$ over a field $K$ is the set of linear combinations $a_{1}\\overrightarrow{\\mathbf{v}}_{1}+\\dots+a_{n}\\overrightarrow{\\mathbf{v}}_{n}$ with $a_{i}\\in K$ . It is denoted $\\operatorname{Sp}(\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{%\\mathbf{v}}_{n})$ . More generally, the span of a set $S$ (not necessarily finite) of vectors is the collection of all (finite) linear combinations of elements of $S$ . The span of the empty set is defined to be the singleton consisting of the zero vector $\\overrightarrow{\\mathbf{0}}$ .

For example, the standard basis vectors $\\hat{\\imath}$ and $\\hat{\\jmath}$ span $\\mathbb{R}^{2}$ because every vector of $\\mathbb{R}^{2}$ can be represented as a linear combination of $\\hat{\\imath}$ and $\\hat{\\jmath}$ .

$\\operatorname{Sp}(\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{%\\mathbf{v}}_{n})$ is a subspace of $V$ and is the smallest subspace containing $\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{\\mathbf{v}}_{n}$ .

Span is both a noun and a verb; a set of vectors can span a vector space, and a vector can be in the span of a set of vectors.

Checking span: To see whether a vector is in the span of other vectors, one can set up an augmented matrix, since if $\\overrightarrow{\\mathbf{u}}$ is in the span of $\\overrightarrow{\\mathbf{v}}_{1},\\overrightarrow{\\mathbf{v}}_{2}$ , then $\\overrightarrow{\\mathbf{u}}=x_{1}\\overrightarrow{\\mathbf{v}}_{1}+x_{2}%\\overrightarrow{\\mathbf{v}}_{2}$ . This is a system of linear equations. Thus, if it has a solution, $\\overrightarrow{\\mathbf{u}}$ is in the span of $\\overrightarrow{\\mathbf{v}}_{1},\\overrightarrow{\\mathbf{v}}_{2}$ . Note that the solution does not have to be unique for $\\overrightarrow{\\mathbf{u}}$ to be in the span.

To see whether a set of vectors spans a vector space, you need to check that there are at least as many linearly independent vectors as the dimension of the space. For example, it can be shown that in $\\mathbb{R}^{n}$ , $n+1$ vectors are never linearly independent, and $n-1$ vectors never span.

Remark. We can define the concept of span also for a module $M$ over a ring $R$ .Given a subset $X\\subset M$ we define the module generated by $X$ as the set of all finite linear combinations of elements of $X$ .Be aware that in general there does not exist a linearly independent subset which generates the whole module, i.e. there does not have to exist a basis.Also, even if $M$ is generated by $n$ elements, it is in general not true that any other set of $n$ linearly independent elements of $M$ spans $M$ .For example $\\mathbb{Z}$ is generated by $1$ as a $\\mathbb{Z}$ -module but not by $2$ .

单词	Span
释义	span The span of a set of vectors $\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{\\mathbf{v_{n}}}$ of a vector space $V$ over a field $K$ is the set of linear combinations $a_{1}\\overrightarrow{\\mathbf{v}}_{1}+\\dots+a_{n}\\overrightarrow{\\mathbf{v}}_{n}$ with $a_{i}\\in K$ . It is denoted $\\operatorname{Sp}(\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{%\\mathbf{v}}_{n})$ . More generally, the span of a set $S$ (not necessarily finite) of vectors is the collection of all (finite) linear combinations of elements of $S$ . The span of the empty set is defined to be the singleton consisting of the zero vector $\\overrightarrow{\\mathbf{0}}$ . For example, the standard basis vectors $\\hat{\\imath}$ and $\\hat{\\jmath}$ span $\\mathbb{R}^{2}$ because every vector of $\\mathbb{R}^{2}$ can be represented as a linear combination of $\\hat{\\imath}$ and $\\hat{\\jmath}$ . $\\operatorname{Sp}(\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{%\\mathbf{v}}_{n})$ is a subspace of $V$ and is the smallest subspace containing $\\overrightarrow{\\mathbf{v}}_{1},\\dots,\\overrightarrow{\\mathbf{v}}_{n}$ . Span is both a noun and a verb; a set of vectors can span a vector space, and a vector can be in the span of a set of vectors. Checking span: To see whether a vector is in the span of other vectors, one can set up an augmented matrix, since if $\\overrightarrow{\\mathbf{u}}$ is in the span of $\\overrightarrow{\\mathbf{v}}_{1},\\overrightarrow{\\mathbf{v}}_{2}$ , then $\\overrightarrow{\\mathbf{u}}=x_{1}\\overrightarrow{\\mathbf{v}}_{1}+x_{2}%\\overrightarrow{\\mathbf{v}}_{2}$ . This is a system of linear equations. Thus, if it has a solution, $\\overrightarrow{\\mathbf{u}}$ is in the span of $\\overrightarrow{\\mathbf{v}}_{1},\\overrightarrow{\\mathbf{v}}_{2}$ . Note that the solution does not have to be unique for $\\overrightarrow{\\mathbf{u}}$ to be in the span. To see whether a set of vectors spans a vector space, you need to check that there are at least as many linearly independent vectors as the dimension of the space. For example, it can be shown that in $\\mathbb{R}^{n}$ , $n+1$ vectors are never linearly independent, and $n-1$ vectors never span. Remark. We can define the concept of span also for a module $M$ over a ring $R$ .Given a subset $X\\subset M$ we define the module generated by $X$ as the set of all finite linear combinations of elements of $X$ .Be aware that in general there does not exist a linearly independent subset which generates the whole module, i.e. there does not have to exist a basis.Also, even if $M$ is generated by $n$ elements, it is in general not true that any other set of $n$ linearly independent elements of $M$ spans $M$ .For example $\\mathbb{Z}$ is generated by $1$ as a $\\mathbb{Z}$ -module but not by $2$ .
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