Update EV6 class activities to clarify spanning R^n and spanning subspace#933
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Update EV6 class activities to clarify spanning R^n and spanning subspace#933
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StevenClontz
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Feb 18, 2026
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Contributor
Perhaps we should await your reflection on its implementation before reviewing then. |
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Yeah I meant to set this as draft until that happened. But I thought it went quite well. I went from no teams realizing that the vectors still span the subspace, to half of the teams making that connection. So the conversation was much better. I don't think I have any further tweaks other than a typo or two I'll address now. |
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StevenClontz
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Feb 18, 2026
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| Similarly, | ||
| <me> | ||
| \setList{ | ||
| \left[\begin{array}{c}1\\2\\3\end{array}\right], | ||
| \left[\begin{array}{c}-2\\0\\5\end{array}\right] | ||
| } | ||
| \text{ and } | ||
| \setList{ | ||
| \left[\begin{array}{c}-1\\2\\8\end{array}\right], | ||
| \left[\begin{array}{c}0\\4\\11\end{array}\right] | ||
| } | ||
| </me> | ||
| are both valid bases for the same planar subspace of <m>\IR^3</m>, | ||
| and they both contain two vectors. |
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Actually here's one more hiccup I fixed on the fly in both classes that I'll fix here now: we emphasize that every basis for
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StevenClontz
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Feb 18, 2026
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StevenClontz
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Feb 18, 2026
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| <definition> | ||
| <statement> | ||
| <p> | ||
| Any non-trivial real vector space has infinitely-many different bases, but all | ||
| the bases for a given vector space are exactly the same size. | ||
| </p> | ||
| <p> | ||
| For example, | ||
| <me> | ||
| \setList{\vec e_1,\vec e_2,\vec e_3} | ||
| \text{ and } | ||
| \setList{ | ||
| \left[\begin{array}{c}1\\0\\0\end{array}\right], | ||
| \left[\begin{array}{c}0\\1\\0\end{array}\right], | ||
| \left[\begin{array}{c}1\\1\\1\end{array}\right] | ||
| } | ||
| \text{ and } | ||
| \setList{ | ||
| \left[\begin{array}{c}1\\0\\-3\end{array}\right], | ||
| \left[\begin{array}{c}2\\-2\\1\end{array}\right], | ||
| \left[\begin{array}{c}3\\-2\\5\end{array}\right] | ||
| } | ||
| </me> | ||
| are all valid bases for <m>\IR^3</m>, and they all contain three vectors. | ||
| </p> | ||
| <p> | ||
| Similarly, | ||
| <me> | ||
| \setList{ | ||
| \left[\begin{array}{c}1\\2\\3\end{array}\right], | ||
| \left[\begin{array}{c}-2\\0\\5\end{array}\right] | ||
| } | ||
| \text{ and } | ||
| \setList{ | ||
| \left[\begin{array}{c}-1\\2\\8\end{array}\right], | ||
| \left[\begin{array}{c}0\\4\\11\end{array}\right] | ||
| } | ||
| </me> | ||
| are both valid bases for the same planar subspace of <m>\IR^3</m>, | ||
| and they both contain two vectors. | ||
| </p> | ||
| </statement> | ||
| </fact> | ||
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| <definition> | ||
| <statement> | ||
| <p> | ||
| The <term>dimension</term> of a vector space or subspace is equal to the size | ||
| of any basis for the vector space. | ||
| So we say the <term>dimension</term> of a vector space or subspace is equal to the | ||
| size of any basis for the vector space. |
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Rather than having two very similar blocks, I've combined the observation that every basis has the same size into the definition of dimension.
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Okay now I'm really ready |
jkostiuk
requested changes
Feb 18, 2026
Co-authored-by: jkostiuk <kostiuk.jordan@gmail.com>
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jkostiuk
approved these changes
Feb 18, 2026
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This is basically the explanation I came up with on the fly today; will try it in my 11am class in a few minutes...