Hello everyone, I found the following statement in my lecture but I don't see why exactly it's true. Any help would be a lot appreciated.
Statement:
I found something maybe useful the following proposition :
Thanks in advance for your help. (It's my first time here tell me if anything is wrong with my post)
Statement:
[MATH]E,F,G,H[/MATH] vector space ( finite-dimensional )
Let [MATH]f \in \mathcal L(E,G)[/MATH] and [MATH]g \in \mathcal L(F,H)[/MATH] ( s.t [MATH]\mathcal L(E,G)[/MATH] is the linear mapping [MATH]E \rightarrow G[/MATH]) .
So we can say that the image of [MATH]f\otimes g \in \mathcal L(E\otimes F,G\otimes H) [/MATH] and is [MATH]f(E)\otimes g(F)[/MATH]
My attempts :Let [MATH]f \in \mathcal L(E,G)[/MATH] and [MATH]g \in \mathcal L(F,H)[/MATH] ( s.t [MATH]\mathcal L(E,G)[/MATH] is the linear mapping [MATH]E \rightarrow G[/MATH]) .
So we can say that the image of [MATH]f\otimes g \in \mathcal L(E\otimes F,G\otimes H) [/MATH] and is [MATH]f(E)\otimes g(F)[/MATH]
I found something maybe useful the following proposition :
[MATH]\mathcal L(E,G) \otimes \mathcal L(F,H) \simeq \mathcal L(E\otimes F,G\otimes H)[/MATH]
Is it enough to assure that [MATH]f\otimes g \in \mathcal L(E\otimes F,G\otimes H) [/MATH] , but I still have no clue how come the image is [MATH]f(E)\otimes g(F)[/MATH].Thanks in advance for your help. (It's my first time here tell me if anything is wrong with my post)
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