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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 ... ...

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 ...

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