Tensor Algebras - Cooperstein Theorem 10.8

[Math Amateur]

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...

I am focused on Section 10.3 The Tensor Algebra ... ...

I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a solution to a UMP ... the theorem is preliminary to tensor algebras ...

I am struggling to understand how the function \(\displaystyle G\) as defined in the proof actually gives us \(\displaystyle G \circ \epsilon_i = g_i\) ... ... if I see the explicit mechanics of this I may understand the functions involved better ... and hence the whole theorem better ... Theorem 10.8 (plus some necessary definitions and explanations) reads as follows:View attachment 5528In the above we read the following:" ... ... Then define\(\displaystyle G(f) = \sum_{j = 1}^t g_{i_j} (f(i_j)) \)We leave it to the reader to show that this is a linear transformation and if G exists then it must be defined in this way, that is, it is unique. ... ... "Can someone please help me to ...

(1) demonstrate explicitly, clearly and in detail that \(\displaystyle G(f) = \sum_{j = 1}^t g_{i_j} (f(i_j)) \) satisfies \(\displaystyle G \circ \epsilon_i = g_i\) (if I understand the detail of this then I may well understand the functions involved better, and in turn, understand the theorem better ...)(2) show that \(\displaystyle G\) is a linear transformation and, further, that if \(\displaystyle G\) exists then it must be defined in this way, that is, it is unique.
Hope that someone can help ...

Peter

 

[Deveno]

I don't know if I will get all the way through this in one sitting, so I'll try to touch on the role of the basic players in this drama.

First of all the $f$'s are kind of confusing. What we are essentially talking about is *finite* $n$-tuples where each "coordinate" is a vector (the $i$-th coordinate is a vector in the $i$-th member of the family).

We add these $n$-tuples by adding the matching "coordinates" in their respective vector spaces.

I'm personally not very fond of this characterization, because I think it obscures that we are just conducting ordinary vector operations in "parallel spaces" that don't communicate with each other. The thing is, you almost have to use this way of describing things when you have an uncountable indexing set $I$, but for a finite number of "factors" (summands) it's more convenient to write the elements of $\bigoplus\limits_{i \in I} V_i$ as regular $n$-tuples.

The reason this often *isn't* done, is that when we have an INFINITE number of summands, we get a SMALLER vector space taking the $f$ with finite support, than the "larger" vector space allowing "any number of coordinates". For example, the direct sum of a countable number of copies of a field is isomorphic to the vector space $F[x]$ considered just as a vector space, whereas allowing "infinite coefficients" is instead isomorphic to $F[[x]]$, the vector space of power series in $F$ (a polynomial coefficient tuple eventually "stops", whereas a power series coefficient tuple may go on indefinitely).

The $\epsilon_i$ can be thought of as "inclusion mappings", where we send each $V_i$ to "its proper place" in the direct sum (much like including the $x$- or $y$-axis in the plane).

So the basic idea with $G$ is we use it to "combine" the family of maps $g_i: V_i \to W$ into "one big map" which you can think of as:

$\sum\limits_i g_i$ (this isn't "quite correct" because there's a domain mis-match, the $g_i$ each have different domains, so we can't add them together "point-wise", but we can use the $\epsilon_i$ to put them all together "where they belong" in the direct sum, and then adding them "makes sense").

For example, if we want to "join" two copies of $\Bbb R$ together, so we can add $x$- and $y$-coordinates "all at once", we can't just write:

$x+y \in \Bbb R^2$, this makes no sense.

So what we have to do is first send:

$x \mapsto (x,0)$
$y \mapsto (0,y)$

And then we can do this:

$(x,0) + (0,y) = (x,y) \in \Bbb R^2$ which DOES make sense.

Now imagine you had some linear maps:

$f: \Bbb R \to W$
$g: \Bbb R \to W$

for some vector space $W$ (it might be a function space, and $f,g$ might represent FAMILIES of functions parameterized by a single variable-for example we might have:

$f(t) = t\cos(x)$, so that for each $t$ we get a "different kind of cosine").

We might want to "combine" these into a single function: $h: \Bbb R^2 \to W$.

We could do this by defining $h(t,u) = f(t) + g(u)$.

Since $f,g$ are linear, we get: $h(t,0) = f(t)$, and $h(0,u) = g(u)$.

Often you see things like: $h = f+g$, which is somewhat incorrect, the notation $f\oplus g$ is better.

 

[Math Amateur]

I don't know if I will get all the way through this in one sitting, so I'll try to touch on the role of the basic players in this drama.

First of all the $f$'s are kind of confusing. What we are essentially talking about is *finite* $n$-tuples where each "coordinate" is a vector (the $i$-th coordinate is a vector in the $i$-th member of the family).

We add these $n$-tuples by adding the matching "coordinates" in their respective vector spaces.

I'm personally not very fond of this characterization, because I think it obscures that we are just conducting ordinary vector operations in "parallel spaces" that don't communicate with each other. The thing is, you almost have to use this way of describing things when you have an uncountable indexing set $I$, but for a finite number of "factors" (summands) it's more convenient to write the elements of $\bigoplus\limits_{i \in I} V_i$ as regular $n$-tuples.

The reason this often *isn't* done, is that when we have an INFINITE number of summands, we get a SMALLER vector space taking the $f$ with finite support, than the "larger" vector space allowing "any number of coordinates". For example, the direct sum of a countable number of copies of a field is isomorphic to the vector space $F[x]$ considered just as a vector space, whereas allowing "infinite coefficients" is instead isomorphic to $F[[x]]$, the vector space of power series in $F$ (a polynomial coefficient tuple eventually "stops", whereas a power series coefficient tuple may go on indefinitely).

The $\epsilon_i$ can be thought of as "inclusion mappings", where we send each $V_i$ to "its proper place" in the direct sum (much like including the $x$- or $y$-axis in the plane).

So the basic idea with $G$ is we use it to "combine" the family of maps $g_i: V_i \to W$ into "one big map" which you can think of as:

$\sum\limits_i g_i$ (this isn't "quite correct" because there's a domain mis-match, the $g_i$ each have different domains, so we can't add them together "point-wise", but we can use the $\epsilon_i$ to put them all together "where they belong" in the direct sum, and then adding them "makes sense").

For example, if we want to "join" two copies of $\Bbb R$ together, so we can add $x$- and $y$-coordinates "all at once", we can't just write:

$x+y \in \Bbb R^2$, this makes no sense.

So what we have to do is first send:

$x \mapsto (x,0)$
$y \mapsto (0,y)$

And then we can do this:

$(x,0) + (0,y) = (x,y) \in \Bbb R^2$ which DOES make sense.

Now imagine you had some linear maps:

$f: \Bbb R \to W$
$g: \Bbb R \to W$

for some vector space $W$ (it might be a function space, and $f,g$ might represent FAMILIES of functions parameterized by a single variable-for example we might have:

$f(t) = t\cos(x)$, so that for each $t$ we get a "different kind of cosine").

We might want to "combine" these into a single function: $h: \Bbb R^2 \to W$.

We could do this by defining $h(t,u) = f(t) + g(u)$.

Since $f,g$ are linear, we get: $h(t,0) = f(t)$, and $h(0,u) = g(u)$.

Often you see things like: $h = f+g$, which is somewhat incorrect, the notation $f\oplus g$ is better.

Thanks for the above post Deveno ... I really needed help on this Theorem ...

You seem to be arguing that Cooperstein is wrong in writing G as \(\displaystyle G(f) = \sum_{j = 1}^t g_{i_j} (f(i_j)) \)and that Cooperstein should be writing \(\displaystyle G(f) = \bigoplus_{i \in I} g_{i_j} (f(i_j)) \)Is that right?

Peter

 

[Math Amateur]

I don't know if I will get all the way through this in one sitting, so I'll try to touch on the role of the basic players in this drama.

First of all the $f$'s are kind of confusing. What we are essentially talking about is *finite* $n$-tuples where each "coordinate" is a vector (the $i$-th coordinate is a vector in the $i$-th member of the family).

We add these $n$-tuples by adding the matching "coordinates" in their respective vector spaces.

I'm personally not very fond of this characterization, because I think it obscures that we are just conducting ordinary vector operations in "parallel spaces" that don't communicate with each other. The thing is, you almost have to use this way of describing things when you have an uncountable indexing set $I$, but for a finite number of "factors" (summands) it's more convenient to write the elements of $\bigoplus\limits_{i \in I} V_i$ as regular $n$-tuples.

The reason this often *isn't* done, is that when we have an INFINITE number of summands, we get a SMALLER vector space taking the $f$ with finite support, than the "larger" vector space allowing "any number of coordinates". For example, the direct sum of a countable number of copies of a field is isomorphic to the vector space $F[x]$ considered just as a vector space, whereas allowing "infinite coefficients" is instead isomorphic to $F[[x]]$, the vector space of power series in $F$ (a polynomial coefficient tuple eventually "stops", whereas a power series coefficient tuple may go on indefinitely).

The $\epsilon_i$ can be thought of as "inclusion mappings", where we send each $V_i$ to "its proper place" in the direct sum (much like including the $x$- or $y$-axis in the plane).

So the basic idea with $G$ is we use it to "combine" the family of maps $g_i: V_i \to W$ into "one big map" which you can think of as:

$\sum\limits_i g_i$ (this isn't "quite correct" because there's a domain mis-match, the $g_i$ each have different domains, so we can't add them together "point-wise", but we can use the $\epsilon_i$ to put them all together "where they belong" in the direct sum, and then adding them "makes sense").

For example, if we want to "join" two copies of $\Bbb R$ together, so we can add $x$- and $y$-coordinates "all at once", we can't just write:

$x+y \in \Bbb R^2$, this makes no sense.

So what we have to do is first send:

$x \mapsto (x,0)$
$y \mapsto (0,y)$

And then we can do this:

$(x,0) + (0,y) = (x,y) \in \Bbb R^2$ which DOES make sense.

Now imagine you had some linear maps:

$f: \Bbb R \to W$
$g: \Bbb R \to W$

for some vector space $W$ (it might be a function space, and $f,g$ might represent FAMILIES of functions parameterized by a single variable-for example we might have:

$f(t) = t\cos(x)$, so that for each $t$ we get a "different kind of cosine").

We might want to "combine" these into a single function: $h: \Bbb R^2 \to W$.

We could do this by defining $h(t,u) = f(t) + g(u)$.

Since $f,g$ are linear, we get: $h(t,0) = f(t)$, and $h(0,u) = g(u)$.

Often you see things like: $h = f+g$, which is somewhat incorrect, the notation $f\oplus g$ is better.

Hi Deveno ... while we are on the topic of Cooperstein's notation ... I wonder if you can clarify Cooperstein's use of the notation \(\displaystyle \epsilon (v) (j)\) ...

My thoughts are, firstly since we have \(\displaystyle \epsilon\) as a map from \(\displaystyle V_i\) to \(\displaystyle \bigoplus_{i \in I} V_i \) it might be better to write \(\displaystyle \epsilon (v_i)\) ...

But what does Cooperstein mean by \(\displaystyle \epsilon (v_i) (j)\) ...

I can only think that he means the \(\displaystyle j\)th coordinate ( or jth entry) in the (possibly) infinite sequence:\(\displaystyle (0_1, 0_2, \ ... \ ... \ , 0_{i-1}, v_i, 0_{i +1}, \ ... \ ... \ ... \ ... )\)Can you comment on my interpretation of Cooperstein's terminology ...

Hope you can help ...

Peter

 

[Deveno]

Hi Deveno ... while we are on the topic of Cooperstein's notation ... I wonder if you can clarify Cooperstein's use of the notation \(\displaystyle \epsilon (v) (j)\) ...

My thoughts are, firstly since we have \(\displaystyle \epsilon\) as a map from \(\displaystyle V_i\) to \(\displaystyle \bigoplus_{i \in I} V_i \) it might be better to write \(\displaystyle \epsilon (v_i)\) ...

But what does Cooperstein mean by \(\displaystyle \epsilon (v_i) (j)\) ...

I can only think that he means the \(\displaystyle j\)th coordinate ( or jth entry) in the (possibly) infinite sequence:\(\displaystyle (0_1, 0_2, \ ... \ ... \ , 0_{i-1}, v_i, 0_{i +1}, \ ... \ ... \ ... \ ... )\)Can you comment on my interpretation of Cooperstein's terminology ...

Hope you can help ...

Peter

It's ok to write

$G(f) = \sum\limits_{j=1}^tg_{i_j}(f(i_j))$

insofar as that goes, because the images are all in the "same space", my quibble is with the importance he places on $f$, which I find to be a notational distraction. I feel that it's just better to write an element of $\bigoplus\limits_i V_i$ as:

$(v_1,\dots,v_k,\dots)$ where every coordinate after the $k$-th is a 0-vector (just as with polynomials, we may have a different $k$ for different elements in the direct sum).

Of course, this implies the $V_i$ can be linearly ordered, which might not be true. As I mentioned before, this isn't an issue until the indexing set $I$ is infinite.

That's the whole point of the support of each $f$ being finite.

In most "concrete" examples, the $f$'s are suppressed.

The reason Cooperstein writes $\epsilon_i(v)(j)$, is because he has committed to writing the vectors that live in the direct sum as certain functions that have finite support. All they are, are embedding functions of the individual factors in the direct sum (they are injective, in fact).

You can't just write $\epsilon(v_i)$ because you have a *different* injection map for each factor. You could write something like:

$\epsilon_j(v_i)$, if it weren't for those pesky $f$'s.

 

[Math Amateur]

It's ok to write

$G(f) = \sum\limits_{j=1}^tg_{i_j}(f(i_j))$

insofar as that goes, because the images are all in the "same space", my quibble is with the importance he places on $f$, which I find to be a notational distraction. I feel that it's just better to write an element of $\bigoplus\limits_i V_i$ as:

$(v_1,\dots,v_k,\dots)$ where every coordinate after the $k$-th is a 0-vector (just as with polynomials, we may have a different $k$ for different elements in the direct sum).

Of course, this implies the $V_i$ can be linearly ordered, which might not be true. As I mentioned before, this isn't an issue until the indexing set $I$ is infinite.

That's the whole point of the support of each $f$ being finite.

In most "concrete" examples, the $f$'s are suppressed.

The reason Cooperstein writes $\epsilon_i(v)(j)$, is because he has committed to writing the vectors that live in the direct sum as certain functions that have finite support. All they are, are embedding functions of the individual factors in the direct sum (they are injective, in fact).

You can't just write $\epsilon(v_i)$ because you have a *different* injection map for each factor. You could write something like:

$\epsilon_j(v_i)$, if it weren't for those pesky $f$'s.

Hi Deveno,

I must be missing something ... since I have no idea what you mean when you write:

"You can't just write $\epsilon(v_i)$ because you have a *different* injection map for each factor. You could write something like:

$\epsilon_j(v_i)$, if it weren't for those pesky $f$'s."Can you explain what you mean in a bit more detail ...

As far as I can see $\epsilon(v_i)$ is quite meaningful ... if taken as an infinite sequence as:

\(\displaystyle \epsilon(v_i) = (0_1, 0_2, \ ... \ ... \ , 0_{i-1}, v_i, 0_{i +1}, \ ... \ ... \ ... \ ... ) \)... ... how do the $f$'s mess with this ... ... ... Can you clarify ...?

Peter

 

[Deveno]

The function that takes an element of $V_1$ and puts it in the first "coordinate" spot is NOT the same function that takes an element of $V_2$ and puts it in the *second* coordinate spot. For one thing, they have different domains.

Although each $\epsilon_i$ does essentially "the same thing", they do it in slightly different ways.

For example, suppose $V_i = \Bbb R^i$.

Then $\epsilon_2$ takes a point in the plane, and that point sits in "the second spot" of our finite sequence of vectors.

But $\epsilon_1$ takes a real number, and puts it in the first spot. These aren't even close to the same function.

 

[Math Amateur]

It's ok to write

$G(f) = \sum\limits_{j=1}^tg_{i_j}(f(i_j))$

insofar as that goes, because the images are all in the "same space", my quibble is with the importance he places on $f$, which I find to be a notational distraction. I feel that it's just better to write an element of $\bigoplus\limits_i V_i$ as:

$(v_1,\dots,v_k,\dots)$ where every coordinate after the $k$-th is a 0-vector (just as with polynomials, we may have a different $k$ for different elements in the direct sum).

Of course, this implies the $V_i$ can be linearly ordered, which might not be true. As I mentioned before, this isn't an issue until the indexing set $I$ is infinite.

That's the whole point of the support of each $f$ being finite.

In most "concrete" examples, the $f$'s are suppressed.

The reason Cooperstein writes $\epsilon_i(v)(j)$, is because he has committed to writing the vectors that live in the direct sum as certain functions that have finite support. All they are, are embedding functions of the individual factors in the direct sum (they are injective, in fact).

You can't just write $\epsilon(v_i)$ because you have a *different* injection map for each factor. You could write something like:

$\epsilon_j(v_i)$, if it weren't for those pesky $f$'s.

Thanks for all your help Deveno ... I am (slowly ... :( ...) getting an understanding of what is going on thanks to your help ...

I really messed up my last question by forgetting the subscript for \(\displaystyle \epsilon_i (v_i)\) and writing \(\displaystyle \epsilon (v_i)\) ...

So ... sorry about that error ...

My question was always about the role of "the pesky \(\displaystyle f\)'s" as you call them ...

Can you explain in more detail what you mean by ...

"You could write something like:

$\epsilon_j(v_i)$, if it weren't for those pesky $f$'s." ?Hope you can clarify ...

Peter

============================================================

*** EDIT ***

To explain my error in the previous post ... when I wrote:

" ... ... As far as I can see $\epsilon(v_i)$ is quite meaningful ... if taken as an infinite sequence as:

\(\displaystyle \epsilon(v_i) = (0_1, 0_2, \ ... \ ... \ , 0_{i-1}, v_i, 0_{i +1}, \ ... \ ... \ ... \ ... ) \)... ... how do the $f$'s mess with this ... ... ... Can you clarify ...?" ... ... ... "

I should have written:" ... ... As far as I can see $\epsilon_i (v_i)$ is quite meaningful ... if taken as an infinite sequence as:

\(\displaystyle \epsilon_i (v_i) = (0_1, 0_2, \ ... \ ... \ , 0_{i-1}, v_i, 0_{i +1}, \ ... \ ... \ ... \ ... ) \)... ... how do the $f$'s mess with this ... ... ... Can you clarify ...? ... ... "
Indeed \(\displaystyle \epsilon_i (v_i)\) is a member of \(\displaystyle \oplus_{i \in I } V_i\) and so we could write

\(\displaystyle f' = \epsilon_i (v_i) \) where \(\displaystyle f' \in \oplus_{ i \in I } V_i\) ...

Indeed, \(\displaystyle f'\) is a member of \(\displaystyle \oplus_{ i \in I } V_i\) with a support of one element \(\displaystyle \{i \}\) ...
*** EDIT 2 ***Just thinking things over regarding $\epsilon_j(v_i)$ ...

... ... ... $\epsilon_j(v_i)$ seems a bit problematic in that \(\displaystyle \epsilon_j\) is a map from \(\displaystyle V_j\) to \(\displaystyle \bigoplus_{i \in I} V_i\) and so takes its values from \(\displaystyle V_j\) ... ... so if \(\displaystyle v_i\) is an element from \(\displaystyle V_i\) then the expression $\epsilon_j(v_i)$ does not seem to make much sense ... ... is that correct?