Tensor Algebras - Cooperstein Example 10.1

[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 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

 

[Deveno]

The direct sum of $\bigoplus\limits_i \mathcal{T}_i(V)$ is created just so we can "add tensors of differing rank".

Let's look at a typical element of $\mathcal{T}_3(V)$ where $V = Fv_0$:

It looks like $c_1v_0 \otimes c_2v_0 \otimes c_3v_0 = c_1c_2c_3 (v_0 \otimes v_0 \otimes v_0)$ by trilinearity.

So when $V$ is one-dimensional, the index basically just keeps track of "how many vectors we're tensoring", and the coefficients are derived from ordinary multiplication.