Page 59 - Curvature and Homology

EXERCISES
and define the n functions ril, ..., zin by

where the a; are n2 constants with non-vanishing determinant, then

It follows that the right side of (1.11.4) vanishes at Po. Consequently,
by (1 9.8) the equations (1.1 1.2) are satisfied.
Incidentally, there exists a geodesic coordinate system in terms of
which (g4j)po = 6;. For, we can find real linear transformations of the
(r2), i = 1, ..., n with constant coefficients so that the fundamental
quadratic form may be expressed as a sum of squares.

EXERCISES

A. The tensor product
Let V and W be vector spaces of dimension n over the field F and denote
by V* and W* the dual spaces of V and W, respectively. Let L(V*, W*; F)
denote the space of bilinear maps of V* x W* into F. This vector space is
defined to he the tensor product of V and W and is denoted by V @ W.
1. Define the map u : V x W + V @ W as follows:
u(v, w) (v*, w*) = <v, v*) (w, w*). Then, u is bilinear and u(V x W)
generates V @ W. Denote u(v, w) by v @ w and call u the natural map. u is
onto but not 1-1.
Hint: To prove that u is onto choose a basis e,, ..., en for V and a basis fl, ..., fn
for W.
2. Let Z be a vector space over F and 8 : V x W4 a bilinear map. Then,
Z
there is a unique linear map 8 : V @I W-+ Z such that 8 . u = 8.

This property characterizes the tensor product as is shown in the following
exercise.