Linear System Theory

Examples Paper 1Functions and Linear Spaces

Due date: October 6, 2014

Exercise 1. (Functions)Consider two sets X and Y and a function f : X Y . Show that:

1. f has a left inverse if and only if it is injective.

2. f has a right inverse if and only if it is surjective.

3. f is invertible if and only if it is bijective.

4. Show that if f is bijective, then all inverses (left-, right-, and two-sided) coincide.

Exercise 2. (Linear spaces)

1. Let S be a set, and F = {f : S R+} the space of functions from S to the (strictly)positive reals. Let the operations : F F F, : R F F be defined asfollows:

[f1 f2](x) = f1(x)f2(x) f1, f2 F, x S[ f ](x) = f(x) R,f F, x S

Show that (F,R,,) is a linear space. Identify the zero-vector.

2. Let S = {a, b}, and letf1(a) = 1, f1(b) = 2

f2(a) = 2, f2(b) = 1

f3(a) = 1, f3(b) = 4

Show that {f1, f2} are linearly independent and that {f1, f3} are linearly dependent.3. Let : F F be defined as follows:

[(f)](x) =f(x) f F, x S

Show that is a linear function over the space F on (F,R,,).

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