Math Amateur
Gold Member
MHB
- 3,920
- 48
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:View attachment 5552
View attachment 5553
View attachment 5554My 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:View attachment 5552
View attachment 5553
View attachment 5554My 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
Last edited: