Let be a field, let be an -vector space, and consider the tensor square . Prove that for all nonzero , if and only if for some .
Let be a basis for . Note then that every simple tensor (hence every element) of can be written in the form . For each , we have an -module injection given by . By the universal property of direct sums, we have an -module homomorphism such that . Now define by . This mapping is well defined since every element of is a finite linear combination of the . Moreover, is clearly bilinear. Thus we have an -module homomorphism such that , where denotes the th natural injection .
We claim that and are mutual inverses. To see this, note that and .
Suppose now that . Let and . Then , so that for all . If , then for all , and since , we have for all . Thus , a contradiction. Thus for some , and we have . Thus and are linearly dependent.
Conversely, suppose . Then .