- #1
Chris L T521
Gold Member
MHB
- 915
- 0
Thanks to those who participated in last week's POTW. I'll just say it's no fun unless more people participate!
This week's problem was again proposed by yours truly (it would be nice if more people proposed some problems! (Smile)).
-----
Problem: Let $\phi:V\rightarrow V$ be a linear operator on a finite-dimensional vector space $V$. Let $k=\text{dim}(\ker \phi)$, and let $d=\text{dim}(V)$. Let $\phi^2:V\otimes V\rightarrow V\otimes V$ be the unique linear operator which satisfies $\phi^2(v_1\otimes v_2) = \phi(v_1)\otimes \phi(v_2)$ for all $v_1, v_2\in V$. Prove that $\text{dim}(\ker\phi^2) = 2dk-k^2$.
-----
I will provide some hints for this week's problem:
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-(POTW)-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
This week's problem was again proposed by yours truly (it would be nice if more people proposed some problems! (Smile)).
-----
Problem: Let $\phi:V\rightarrow V$ be a linear operator on a finite-dimensional vector space $V$. Let $k=\text{dim}(\ker \phi)$, and let $d=\text{dim}(V)$. Let $\phi^2:V\otimes V\rightarrow V\otimes V$ be the unique linear operator which satisfies $\phi^2(v_1\otimes v_2) = \phi(v_1)\otimes \phi(v_2)$ for all $v_1, v_2\in V$. Prove that $\text{dim}(\ker\phi^2) = 2dk-k^2$.
-----
I will provide some hints for this week's problem:
Apply the rank-nullity theorem. Also note that if $V$ and $W$ are finite-dimensional vector spaces and $V\otimes W$ is the tensor product of $V$ with $W$, then $\text{dim}(V\otimes W) = \text{dim}(V)\cdot\text{dim}(W)$.
Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-(POTW)-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!