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Vector Spaces: Cartesian vs Tensor products

  1. Sep 5, 2011 #1
    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
     
  2. jcsd
  3. Sep 5, 2011 #2
    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).
     
  4. Sep 5, 2011 #3
    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?
     
  5. Sep 5, 2011 #4

    mathwonk

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    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.
     
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