This is a question in my textbook: "The hyperplanes H1 and H2 have dimensions p and q, respectively. What is the smallest dimension which the hyperplane H3 must have in order to be sure to contain both H1 and H2?" I reasoned it out like this. A basis for H1 would be x1 + x2 +...+ xp And a basis for H2 would be xp+1 + xp+2 +...+xq So, a hyperplane which would contain all of these vectors would have the basis: x1 + x2 +...+ xp + xp+1 +...+ xq So its dimension would be p+q. However, the answer in the back of my book says the answer is p + q + 1. What is wrong with what I did?