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I am a computer science student and do not have any background about tensors. Recently, I started learning about tensor spaces and tensor products from the text "Tensor spaces and exterior algebra" by Takeo Yokonuma.

I had a question and I understand that its pretty basic - but I would like some help with that.

The author is trying to show that given two vector spaces $V$ and $W$ over a field $k$ and any bilinear map $\phi \in L(V,W,U)$ (EDIT: where $L(V,W,U)$ is the set of all bilinear maps from $V \times W \rightarrow U$ for various vector spaces $U$ over the field $k$.) We consider a particular bilinear mapping $l \colon V \times W \to U_0$ (i) there exists a $k$-linear mapping $F \ \colon U_0 \to U$ such that $\phi = F \ o \ l$. () (ii) $l$ (in the former part) is the mapping from $V \times W$ to $U_0$.

I was going through the proof of part (i) which is where my troubles begin.

Author has already established that the $dim(U_0) \geq mn$ where $dim(V) = n$ and $dim(W) = m$. He proceeds by picking a $U_0$ of dimension exactly $mn$. Thereafter, he takes a basis $\mathcal{G} = {g_1, g_2 \ldots g_{mn}}$ of $U_0$ and observes that the basis $\mathcal{E}$ of $U$ and $\mathcal{F}$ of $W$ are such that we have a one-to-one correspondence between the sets $\mathcal{E} \times \mathcal{F}$ and $\mathcal{G}$.

After this observation, the author lets $\phi \in L(V,W,U)$ for a vector space $U$. Then, he defines $F_{\phi} \colon U_0 \to U$ by

$F_{\phi}(u) = F_{\phi}(\sum_{i,j}\gamma_{i,j} \cdot g(i,j) = \sum_{i,j} \gamma_{i,j} \cdot \phi(i,j))$

and this is what I have trouble believing. I am willing to believe that $F_{\phi} (\gamma_{ij}g(i,j)) = \gamma_{ij} \phi(i,j)$ but I cannot assume that this also distributes linearly - as this is what I want to prove.

Maybe it is easy to see but I would like to have some help here (as I do not see why is this definition for $F_{\phi}$ legitimate)

Hope my question is clear. Please help Thanks -Akash

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What is $U_0$? Having undefined things pop out from nowhere is bad form. I think you mean $U_0 = V \otimes W$? –  kahen Jan 15 '11 at 8:48
    
I second that.... –  user02138 Jan 15 '11 at 9:10
    
I am sorry, I will make that edit –  Akash Kumar Jan 15 '11 at 13:43

1 Answer 1

up vote 3 down vote accepted

Linearity is built into the definition: you first define the map $F_\phi$ on the basis elements, then you extend linearly. I am not sure what you mean by "this is what I have trouble believing". Clearly, if you only define a map on a basis and then extend linearly to linear combinations of the basis elements, you get a linear map. The only thing that you might doubt is that this is well-defined. But in the context of vector spaces there is no such issue, since every element can be uniquely written as a linear combination of the basis elements.

Think of what you do when you define a linear map on a vector space. You just specify what this map does to a basis (that's the whole point behind matrix notation).

By the way, if you tried to do that on an arbitrary generating set of some module, then you would be heading for trouble without appropriate care, so your doubt would be appropriate.

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Thanks Alex. Let me embarrass myself a little more. I am okay with $F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0)) = \gamma_{i_0,j_0}\phi(i_0,j_0)$ and $F_{\phi}(\gamma_{i_1,j_1}g(i_1,j_1)) = \gamma_{i_1,j_1}\phi(i_1,j_1)$ but I am uncomfortable with $F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0) + \gamma_{i_1,j_1}g(i_1,j_1)) = F_{\phi}(\gamma_{i_0,j_0}g(i_0,j_0))+ F_{\phi}(\gamma_{i_1,j_1}g(i_1,j_1))$. It seems that I am assuming that $F_{\phi}$ distributes linearly over the basis elements and I am not sure why is this correct. –  Akash Kumar Jan 15 '11 at 14:10
1  
@Akash You are not assuming it, you are defining $F_\phi$ to distribute linearly. That's how you define it on a sum of elements: to be the sum of images under $F_\phi$ of each of the elements. –  Alex B. Jan 15 '11 at 14:22
    
oh...i understand now. thanks for your help alex. –  Akash Kumar Jan 15 '11 at 15:02

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