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Does $\Bbb R^3=\textrm{span}\ \Bbb R^3$? I would imagine it does considering $\Bbb R^3$ contains all linear equations with three variables and $\textrm{span}\ \Bbb R^3$ contains all the linear combinations of these said equations.

Am I correct?

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up vote 2 down vote accepted

Your conclusion is correct; your argument for it is a bit off. ${\bf R}^3$ doesn't contain any linear equations; it contains ordered triples of real numbers. The set of all linear combinations, with real coefficients, of ordered triples of real numbers, is the set of all ordered triples of real numbers.

More generally, if $V$ is any vector space, then the span of $V$ is $V$.

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One needs to take care when discussing the span of an infinite set of vectors. In particular, $\textrm{span}\ \mathbb{R}^3$ is the set of all finite linear combinations of vectors in $\mathbb{R}^3$ which is, as you say, $\mathbb{R}^3$. Alternatively, you can define the span of a set of vectors in terms of an intersection of subspaces; this definition allows for both finite and infinite collections of vectors. – Michael Albanese Feb 5 '13 at 4:54

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