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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 ##G## as defined in the proof actually gives us ##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:
In the above we read the following:
" ... ... Then define
##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 ##G(f) = \sum_{j = 1}^t g_{i_j} (f(i_j)) ## satisfies ##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 ##G## is a linear transformation and, further, that if ##G## exists then it must be defined in this way, that is, it is unique.
Hope that someone can help ...
Peter
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 ##G## as defined in the proof actually gives us ##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:
In the above we read the following:
" ... ... Then define
##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 ##G(f) = \sum_{j = 1}^t g_{i_j} (f(i_j)) ## satisfies ##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 ##G## is a linear transformation and, further, that if ##G## exists then it must be defined in this way, that is, it is unique.
Hope that someone can help ...
Peter
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