If
and
are both inner products on a vector space V, then
is also an inner product on V.

Question

If 
  and 
  are both inner products on a vector space V, then
  is also an inner product on V.

Quizplus · Accepted Answer

The new inner product can be obtained by adding the corresponding entries of the two given inner products and then dividing by 2, since the sum of two inner products is also an inner product and scaling an inner product by a positive scalar also results in an inner product. Thus, we have:

11edaba5_20b8_3585_a975_af8e9990af4f_TBMC1039_00 + 11edaba5_20b8_3586_a975_0d3c780c24e5_TBMC1039_00 = 11edaba5_20b8_3587_a975_1d10bcae34dd_TBMC1039_00

(assuming that the vector space V has the same dimension for both inner products)

If and Are Both Inner Products on a Vector Space