Vector Spaces: Cartesian vs Tensor products

[Monte_Carlo]

Hi,

I have a problem understanding the difference between Cartesian product of vector spaces and tensor product. Let V1 and V2 be vector spaces. V1 x V2 is Cartesian product and V1 xc V2 is tensor product (xc for x circled). How many dimensions are in V1 x V2 vs V1 xc V2?

Thanks,

Monte

 

[fscman]

It really depends how you define addition on cartesian products. The usual definition is

$(v_1+v_2,w_1+w_2)=(v_1,w_1)+(v_2,w_2)$

In this case, the cartesian product is usually called a direct sum, written as $V \oplus W$.
If you think about it, this 'product' is more like a sum--for instance, if $v_1,v_2,...v_n$ are a basis for $V$ and $w_1,w_2,...w_m$ are a basis for W, then a basis for $V \oplus W$ is given by
$v_1 \oplus 0, ..., v_n \oplus 0, 0 \oplus w_1, ..., 0 \oplus w_m$, and so the dimension is $n+m$

A tensor product, on the other hand, is actually a product (which can be thought of as a concatenation of two vectors) that obeys the distributive law:

$(v_1+v_2)\otimes (w_1+w_2)=v_1 \otimes w_1 + v_1 \otimes w_2 + v_2 \otimes w_1 + v_2 \otimes w_2$

One basis is
$v_1 \otimes w_1, v_1 \otimes w_2, ..., v_2 \otimes w_1, ..., ..., v_n \otimes w_m$
and the space has dimension mn (as expected of a product).

 

[Monte_Carlo]

I'm having a hard time following because my computer doesn't show the symbols in a standard mathematical notation. Would you be able to refer to some online source with the same information?

 

[mathwonk]

if a,b,c and x,y are bases of V, W then (a,0),(b,0),(c,0),(0,x),(0,y) is a basis of the cartesian product VxW, while (a,x), (b,x),(c,x),(a,y),(b,y),(c,y) is a basis of the tensor product VtensW.

so dimension is additive for cartesian product and multiplicative for tensor product.