Vector space properties In example 1.2.2, the textbook says that the subspace {x E R" : all1 + . . . + andn = 0, where a E R for all i}, is sometimes called a hyperplane. Here, we clarify that it is a hyperplane if it is not all of R". We can extend this to hyperplanes in Fr for any field, by replacing R with F. We write Fs for the field with 5 elements. 3. The vector space F? = {(a, b) : a, b E F5} has 6 distinct hyperplanes. (a) Describe them. (b) Given a point in F', explain how you could tell which hyperplanes it is on. 4. In the vector space F;, the hyperplane S contains all points (x, y, z) with x + 2y + 4z = 0, and the hyperplane T contains all points with 2x + 4y + 3z = 0. (a) What is the intersection Sn T? (b) How many distinct hyperplanes does F; have? Describe them. (c) Given a point in F, explain how you could tell which hyperplanes it is on