# I The Tensor Algebra - Cooperstein, Example 10.1

1. Apr 26, 2016

### 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:

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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2. Apr 26, 2016

### stevendaryl

Staff Emeritus
Since the $\mathcal{T}_k$ are vector spaces, a vector space is closed under multiplication by a constant. So if $v \otimes \ ... \ ... \ \otimes v$ is an element of the vector space, so is $c v \otimes \ ... \ ... \ \otimes v$

3. Apr 26, 2016

### andrewkirk

The key is where the author says 'We will often abuse notation and express $\boldsymbol x$ as a sum of its homogeneous parts rather than as a function from $\mathbb Z_{\geq 0}$'.

I note that this also explains the abuse of notation that caused you concern in your other post from yesterday. I have to retract my criticism of the author that I made there, as the excerpt you include here shows that he does explicitly acknowledge the abuse of notation. I beg your pardon, Mr Cooperstein.

The elements that are being added are functions from $\mathbb Z_{\geq 0}$ to $\bigcup_{k\in\mathbb Z_{\geq 0}}\mathcal T_k$. Because we have the constraint $\boldsymbol x(k)\in\mathcal T_k(V)$ and $\mathcal T_k(V)$ is a vector space (and hence has well-defined addition), we have the following natural definition of addition in $\mathcal T(V)$. For $\boldsymbol x,\boldsymbol y\in\mathcal T(V)$, $\boldsymbol x+\boldsymbol y$ is the function from $\mathbb Z_{\geq 0}$ to $\bigcup_{k\in\mathbb Z_{\geq 0}}\mathcal T_k$ such that, for all $k\in\mathbb Z_{\geq 0}$:
$$(\boldsymbol x+\boldsymbol y)(k)=\boldsymbol x(k)+\boldsymbol y(k)$$

4. Apr 26, 2016

### Staff: Mentor

To get an idea of the tensors you can write vectors in coordinates, say $v = (v_1,...,v_n) , w = (w_1,...,w_n)$.
$\mathcal T_0 = \mathbb{F}, \mathcal T_1 = V$ and $\mathcal T_2$ can be seen as all $(n \times n)$ matrices.
An element $v\otimes w$ then is the rank $1$ matrix
$$v^τ \cdot w = \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix} \cdot (w_1, \cdots, w_n) = \begin{bmatrix} v_1w_1 &&\cdots && v_1w_n \\ \vdots && \cdots && \vdots \\ v_nw_1 && \cdots && v_nw_n\end{bmatrix}$$
and addition together with scalar multiplications of those matrices give you any $(n \times n)$ matrix you want.
The same procedure on $u\otimes v\otimes w$ will deliver a $(n \times n \times n)$ cube of rank $1$ and so on.

You are right that you cannot write the sum $c + v + v \otimes w + u \otimes v \otimes w$ other than this formal sum. However, in each component (the homogeneous parts of degree $0, 1, 2, 3,$ resp.) you can add and (scalar) multiply the elements as I mentioned in the case of rank $1$ matrices.

Another picture is that of a polynomial ring micromass mentioned elsewhere. If you have $ℝ[X,Y]$ you cannot add $X$ and $Y$ other than writing $X+Y$. Nevertheless this doesn't keep you from calculating with polynomials in two variables.

5. Apr 26, 2016

### Math Amateur

Steven, Andrew and Fresh ... thank you for your helpful and clarifying posts ...

I am reflecting on what you have said ...

Thanks again,

Peter