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Does the $\operatorname{span} \{(1, 0, 0), (0, 1, 0)\}= \mathbb{R}^2?$ I was told the span of this set has dimension $2$ but what is the exact span?

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$\mathbf{R}^2$ is the set of all pairs of real numbers, so no, $\mathrm{Vect}(\{(1,0,0),(0,1,0)\})$ is not $\mathbf{R}^2$. It is, however, isomorphic to $\mathbf{R}^2$ (meaning that there exists a bijective linear map between the two).

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  • $\begingroup$ Which basically means you replace $=$ with $\cong$ in your equation. The key is, all spans of two independent vectors in any real vector space are isomorphic to $\mathbb R^2$ $\endgroup$ Commented May 5, 2014 at 3:06

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