I currently self study from the book "A Course in Modern Mathematical Physics" by Peter Szekeres, and I'm currently reading the chapter on tensors, which he defines using the concept of Free Vector Spaces.(adsbygoogle = window.adsbygoogle || []).push({});

He gives a re-definition of Grassmann's algebras introduced in the previous section by using the concept of free algebra. And I had the following problem while reading the text:

(This is a Citation:) "

Let [tex]\mathcal{F}(V)[/tex] be the free associative algebra over a real vector space [tex]V[/tex], and let [tex]\mathcal{S}[/tex] be the ideal generated by all elements of [tex]\mathcal{F}(V)[/tex] of the form [tex]u \otimes T \otimes v + v \otimes T \otimes u[/tex] where [tex]u,v\in V[/tex] and [tex]T\in \mathcal{F}(V)[/tex]. The general element of [tex]\mathcal{S}[/tex] is [tex]S \otimes u \otimes T \otimes v \otimes U + S \otimes v \otimes T \otimes u \otimes U[/tex] where [tex]u,v\in V[/tex] and [tex]S,T,U\in \mathcal{F}(V)[/tex]

"

My question is: Why can every element be expressed in this way?

An Ideal is first of all a Vector Subspace, right? So the sum of any two such "general elements" is supposed to be a general element too.

In other words,how do I prove that:

[tex]S_1 \otimes u_1 \otimes T_1 \otimes v_1 \otimes U_1 + S_1 \otimes v_1 \otimes T_1 \otimes u_1 \otimes U_1 \ \ \ \ + \ \ \ \

S_2 \otimes u_2 \otimes T_2 \otimes v_2 \otimes U_2 + S_2 \otimes v_2 \otimes T_2 \otimes u_2 \otimes U_2 \ \ \ \ = \ \ \ \

S_3 \otimes u_3 \otimes T_1 \otimes v_3 \otimes U_3 + S_3 \otimes v_3 \otimes T_3 \otimes u_3 \otimes U_3[/tex]

?

Thank you for your time.

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# Free Algebras

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