Math 206 Homework #11

Reading

Study the first 5 pages of Section 5.2: Subspaces (so pp 211 through nearly all of p215).
As you need to, continue to study Section 5.1: Real Vector Spaces.
Collect the handout "Sets: Notation and Terminology" from the handouts page. Read the first page and skim the second. Pay particular attention to the meaning and notation for: empty set, intersection of two sets, union of two sets.You won't need this for the HWK, but you will need intersection and union for GP11, and it would be good in any case to become familiar with this notation and terminology.
If you didn't already do so, collect the handout "Subspaces" as well. Read it and keep it handy as you do the problems from 5.2.

Problems

5.1 p208: 6, 16
5.2 p219: 1ad, 2bc, 3ac, 4, 5, 6bcf, 17, 22, 23ab.
In #22, the underlying vector space uses V = C[a,b], which is the set of all real-valued functions on [a,b] that are also continuous on [a,b]. Addition is usual addition of functions. Scalar multiplication of functions is usual scalar multiplication of functions. [So f+g is the function defined by (f+g)(x) = f(x)+g(x), and kf is the function defined by (kf)(x) = k(f(x)). In other words the value at x of the function f+g is the sum of f(x) and g(x). The value at x of the scalar multiple kf is k times f(x). Note that the zero vector for V is the constant function 0, i.e. the function that assigns to each x the value 0.]