- #1
Chris L T521
Gold Member
MHB
- 915
- 0
This will be my last University POTW submission here on MHB; I'm running out of ideas (after doing this for 133 weeks), and I think it's time to get someone fresh in here to do things from now on. It's been a pleasure doing this for over 2.5 years now, and I hope you guys give the person who will be taking my place the same kind of support I've received from you during my time running the University POTW.
Anyway, here's this week's problem!
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Problem: Let $V$ be a Euclidean space and $W$ a subspace of $V$. Show that if $\mathbf{w}_1,\ldots,\mathbf{w}_r$ is a basis for $W$ and $\mathbf{u}_1,\ldots,\mathbf{u}_s$ is a basis for $W^{\perp}$, then $\mathbf{w}_1,\ldots,\mathbf{w}_r, \mathbf{u}_1,\ldots,\mathbf{u}_s$ is a basis for $V$ and that $\dim V = \dim W + \dim W^{\perp}$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
Anyway, here's this week's problem!
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Problem: Let $V$ be a Euclidean space and $W$ a subspace of $V$. Show that if $\mathbf{w}_1,\ldots,\mathbf{w}_r$ is a basis for $W$ and $\mathbf{u}_1,\ldots,\mathbf{u}_s$ is a basis for $W^{\perp}$, then $\mathbf{w}_1,\ldots,\mathbf{w}_r, \mathbf{u}_1,\ldots,\mathbf{u}_s$ is a basis for $V$ and that $\dim V = \dim W + \dim W^{\perp}$.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!