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Alexander.zw (talk | contribs) Add the example of the span of the empty set |
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The set {(1,0,0), (0,1,0), (1,1,0)} is not a spanning set of '''R'''<sup>3</sup>; instead its span is the space of all vectors in '''R'''<sup>3</sup> whose last component is zero. That space (the space of all vectors in '''R'''<sup>3</sup> whose last component is zero) is also spanned by the set {(1,0,0), (0,1,0)}, as (1,1,0) is a linear combination of (1,0,0) and (0,1,0). It does, however, span '''R'''<sup>2</sup>.
The empty set is a spanning set of {(0, 0, 0)} since the empty set is a subset of all possible vector spaces in '''R'''<sup>3</sup>, and {(0, 0, 0)} is the intersection of all of these vector spaces.
The set of functions ''x<sup>n</sup>'' where ''n'' is a non-negative integer spans the space of polynomials.
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