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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 Example 10.1 in Section 10.3 ...
Example 10.1 plus some preliminary definitions reads as follows:
My questions related to Example 10.1 are articulated below ... ...
Question 1
In the above text from Cooperstein we read in Example 1, the following:
" ... ... Then ##\mathcal{T}_k (V) = \{ cv \otimes \ ... \ ... \ \otimes v \  \ c \in \mathbb{F} \}## ... ... "
But ... ##\mathcal{T}_k (V)## is defined by
##\mathcal{T}_k (V) = V \otimes V \otimes V \ ... \ ... \ \otimes V## ... ... ... (1)
( and there are ##k## ##V##'s in the product ... )
... surely then ##\mathcal{T}_k (V) = \{ v \otimes \ ... \ ... \ \otimes v \  \ v \in V \} ##
and not (as shown in Cooperstein Example 10.1 )
##\mathcal{T}_k (V) = \{ cv \otimes \ ... \ ... \ \otimes v \  \ c \in \mathbb{F} \} ##
... can someone please explain why ##\mathcal{T}_k (V)## has the form shown by Cooperstein in Example 10.1 ...
Question 2
Can someone explain how/why the general element of degree 3 is as shown in Example 10.1 ...
Does it make sense to add these elements ... they seem different in nature and form ...
Hope 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 Example 10.1 in Section 10.3 ...
Example 10.1 plus some preliminary definitions reads as follows:
My questions related to Example 10.1 are articulated below ... ...
Question 1
In the above text from Cooperstein we read in Example 1, the following:
" ... ... Then ##\mathcal{T}_k (V) = \{ cv \otimes \ ... \ ... \ \otimes v \  \ c \in \mathbb{F} \}## ... ... "
But ... ##\mathcal{T}_k (V)## is defined by
##\mathcal{T}_k (V) = V \otimes V \otimes V \ ... \ ... \ \otimes V## ... ... ... (1)
( and there are ##k## ##V##'s in the product ... )
... surely then ##\mathcal{T}_k (V) = \{ v \otimes \ ... \ ... \ \otimes v \  \ v \in V \} ##
and not (as shown in Cooperstein Example 10.1 )
##\mathcal{T}_k (V) = \{ cv \otimes \ ... \ ... \ \otimes v \  \ c \in \mathbb{F} \} ##
... can someone please explain why ##\mathcal{T}_k (V)## has the form shown by Cooperstein in Example 10.1 ...
Question 2
Can someone explain how/why the general element of degree 3 is as shown in Example 10.1 ...
Does it make sense to add these elements ... they seem different in nature and form ...
Hope someone can help ...
Peter
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