Okay so in my lecture notes the product of two subspaces A and B is defined as the span of dot products for each pair of spanning vectors in A and B. Here in Exercise 6.45 using this definition we get A.B as the span of the four unit indicator vectors. This means we have A.B = R^4, correct?
Well, in 6.46 we are to demonstrate that the product of subspaces is not necessarily itself a subspace, using 6.45... Huh? If A.B is R^4 then surely it is a subspace. e_1 + e_4 = (1,0,0,1) is absolutely in A.B as it was defined in 6.45. But the next part shows that no two vectors from A and B can produce the dot product of (1,0,0,1) so surely A.B can't be R^4... So what's going on here?