1.

In elementary algebra and calculus courses, it is often said that f : R ? R is linear if and only if it has the form f( x) = ax + b, where a and b are constants. Does this agree with Definition 3.9? If not, what is the form of a linear function f : R ? R?

2.

Show explicitly that f : R ? R defined by f( x) = ? is not linear.

3.

For each of the following sets of functions, determine whether or not it is a vector space. (Define addition and scalar multiplication in the obvious way.) If it is not, state what property of a vector space fails to hold.

1. { f ? C[0, 1] : f(0) = 0}

2. { f ? C[0, 1] : f(0) = 1}

3. { f ? C[0, 1] : ? ¹ ₀ f( x) dx = 0}

4. , the set of all polynomials of degree n or less.

5. The set of all polynomials of degree exactly n.

4.

Prove that the differential operator L : C ¹[ a,b] ? C[ a,b] defined by

is not a linear operator.

5.

Prove that the differential operator L : C ¹[ a,b] ? C[ a,b] defined by

is not a linear operator.