- #1
klackity
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Tell me if this is true:
We are given vector spaces V1, V2, ..., Vn of dimensions d1, d2, ..., dn respectively.
Let V = V1 [tex]\otimes[/tex] V2 [tex]\otimes[/tex] ... [tex]\otimes[/tex] Vn
Claim: Any element v [tex]\in[/tex] V can be represented in the following form:
[tex]\sum[/tex]i=1...R (v1,i [tex]\otimes[/tex] ... [tex]\otimes[/tex] vn,i)
Where R = - MAX {dj} + [tex]\sum[/tex]j=1..n dj
And where vj,i [tex]\in[/tex] Vj.
In other words, there is an upper bound of R on the number of "elementary" tensors wi needed needed to represent a particular tensor v in V (where an "elementary" tensor wi is one which can be written in the form wi = v1 [tex]\otimes[/tex] v2 [tex]\otimes[/tex] ... vn, where vj [tex]\in[/tex] Vj). The upper bound R is simply the sum of all the dj except for the dj0 which is maximal.
Is this true?
We are given vector spaces V1, V2, ..., Vn of dimensions d1, d2, ..., dn respectively.
Let V = V1 [tex]\otimes[/tex] V2 [tex]\otimes[/tex] ... [tex]\otimes[/tex] Vn
Claim: Any element v [tex]\in[/tex] V can be represented in the following form:
[tex]\sum[/tex]i=1...R (v1,i [tex]\otimes[/tex] ... [tex]\otimes[/tex] vn,i)
Where R = - MAX {dj} + [tex]\sum[/tex]j=1..n dj
And where vj,i [tex]\in[/tex] Vj.
In other words, there is an upper bound of R on the number of "elementary" tensors wi needed needed to represent a particular tensor v in V (where an "elementary" tensor wi is one which can be written in the form wi = v1 [tex]\otimes[/tex] v2 [tex]\otimes[/tex] ... vn, where vj [tex]\in[/tex] Vj). The upper bound R is simply the sum of all the dj except for the dj0 which is maximal.
Is this true?