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

[itex](v_1+v_2,w_1+w_2)=(v_1,w_1)+(v_2,w_2)[/itex]

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

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:

[itex](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 [/itex]

One basis is
[itex]v_1 \otimes w_1, v_1 \otimes w_2, ..., v_2 \otimes w_1, ..., ..., v_n \otimes w_m[/itex]
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.

 

[blue_raver22]

Hi Monte,

Great question! The main difference between Cartesian product and tensor product lies in their definitions and properties.

Cartesian product, denoted by V1 x V2, is a set of all possible ordered pairs (v1, v2) where v1 is an element of V1 and v2 is an element of V2. In other words, the elements of the Cartesian product are all possible combinations of elements from V1 and V2. For example, if V1 = {1, 2} and V2 = {3, 4}, then V1 x V2 would be {(1,3), (1,4), (2,3), (2,4)}. The dimension of V1 x V2 would be the product of the dimensions of V1 and V2, which in this case would be 2 x 2 = 4 dimensions.

On the other hand, tensor product, denoted by V1 ⊗ V2, is a mathematical operation that combines two vector spaces to create a new, larger vector space. It is defined as a set of all possible linear combinations of elements from V1 and V2. In other words, the elements of the tensor product are all possible combinations of vectors from V1 and V2, multiplied by scalars. The dimension of V1 ⊗ V2 is the product of the dimensions of V1 and V2, but it also includes additional dimensions to account for the scalars. Using the same example as before, the dimension of V1 ⊗ V2 would be 2 x 2 + 2 = 6 dimensions.

In summary, the main difference between Cartesian product and tensor product is that Cartesian product is a set of elements, while tensor product is a new vector space. This results in a difference in dimensions, with tensor product having more dimensions due to the inclusion of scalars. I hope this helps clarify the difference between the two concepts.

Best,