Showing injectivity directly would involve somehow showing that there are no non-trivial relationships between formula_134 and formula_139 for formula_140, which seems daunting. However, we know that there is a bilinear map formula_141 given by multiplying the coordinates together, and the universal property of the tensor product then furnishes a map of vector spaces formula_142 which maps formula_134 to formula_136, and hence is an inverse of the previously constructed homomorphism, immediately implying the desired result. Note that, a priori, it is not even clear that this inverse map is well-defined, but the universal property and associated bilinear map together imply this is the case.