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Lecture 14
Andrei Antonenko
March 05, 2003
1 Functions
In previous lectures we worked with algebraic structures — sets with operations defined on them. Now we will consider another important thing in mathematics — functions. Let A and B be 2 sets. Function f from A to B can be considered as a rule, which allows us to get an element from B for any element from A. The notation for a function from the set A to the set B is: f : A → B. Set A is called the domain of a function f. We will often use the following notation: x 7 → f (x), which denotes that x maps to f (x), i.e. applying f to x we get f (x). Now let’s consider any element x from A. Then f (x) ∈ B is called the image of x. Moreover we can consider the subset A′^ ⊂ A. Then by f (A′) we will denote the set which contains images of all the elements from A′^ and it will be called the image of A′. Let’s consider any subset in B, say, B′^ ∈ B. Then by f −^1 (B′) we will denote all elements from A, whose images are in B′. f −^1 (B′) will be called the inverse image of preimage of B′.
Example 1.1. Consider the function f (x) = x^2. This function is defined for any real number, and maps them to nonnegative real numbers. If R+ denotes positive numbers, then
f : R → R+ ∪ { 0 },
where R+ ∪ { 0 } is the set of all nonnegative numbers. Image of the set [− 2 , 2] is the set [0, 4] = f ([− 2 , 2]). Inverse image of the set [4, 25] is the set [− 5 , −2] ∪ [2, 5] = f −^1 ([4, 25]).
Consider another important notion — composition of functions. Let f : A → B, g : B → C are 2 functions. Then composition of f and g is the function which is denoted by g ◦ f such that g ◦ f : A → C, and (g ◦ f )(x) = g(f (x)).
We state here the obvious property of compositions: if f : A → B, g : B → C, and h : C → D, then h ◦ (g ◦ f ) = (h ◦ g) ◦ f.
Example 1.2. Consider f : R^2 → R such that (x, y) 7 → xy, and g : R → R such that x 7 → x^3. Then g ◦ f : R^2 → R and (g ◦ f )(x, y) = g(f (x, y)) = g(xy) = (xy)^3.
Now, let A be a nonempty set. The function f : A → A such that for any x ∈ A we have f (x) = x is called the identity function. It will be denoted by I: I(x) = x. Now let f : A → B. Function g : B → A is called the inverse function for f if
(f ◦ g)(x) = x ∀x ∈ B, and (g ◦ f )(x) = x ∀x ∈ A.
Example 1.3. Let f : R → R+ ∪ { 0 } such that f (x) = x^2. Then g(x) = √x is not an inverse: let’s take x = − 2 , then − 2 7 −→f 4 7 −→g 2. But if we consider f (x) = x^2 only for nonnegative numbers, then g(x) =
x will be the inverse.
All these definitions are basic definitions of mathematics, and are not specific for linear algebra.
2 Linear functions
Now, we will consider a class of functions, which is specific to linear algebra. Let V and U are vector spaces.
Definition 2.1. Function f : V → U is called a linear function if the following 2 conditions are satisfied:
- For any vectors v and w from V
f (v + w) = f (v) + f (w)
- For any vector v ∈ V and for any number k ∈ R
f (kv) = kf (v)
Now we’ll give some examples of linear functions.
Example 2.2. The identity function is a linear function. It is easy to see:
I(x + y) = x + y = I(x) + I(y) I(kx) = kx = kI(x)
Example 2.3. Let’s consider the zero-function — function f : V → V such that for any v ∈ V f (v) = 0. This function is obviously a linear function.
Now we’ll state very easy result about linear functions.
Lemma 2.7. If f is a linear function, then f ( 0 ) = 0.
Proof. Let k 6 = 0. Then f (k 0 ) = kf ( 0 ). Moreover, since k 0 = 0 , then f (k 0 ) = f ( 0 ). So, comparing these two equalities, we have that f ( 0 ) = kf ( 0 ), so f ( 0 ) = 0.
Example 2.8. Let f : R^2 → R^2 such that f (x, y) = (x + 1, y + 2). Then this is not linear function, since the image of zero is not zero:
f (0, 0) = (1, 2) 6 = (0, 0).
Actually, all other properties do not hold here as well. For example, let u = (1, 1), and let v = (1, 0). Then f (u) = (2, 3), f (v) = (2, 2), and so, f (u) + f (v) = (4, 5). But f (u + v) = f (2, 1) = (3, 3) 6 = (4, 5).
Now we will consider the most important linear function which will be used widely in the future.
Example 2.9 (Matrix function). Let A be any m × n-matrix. Then we can define the function FA : Rn^ → Rm^ by the following formula: if v ∈ Rn^ then v 7 −→FA Av, where Av is a multiplication of a matrix A by a column-vector (n × 1 -matrix) v.
Consider an example. Let A =
, so FA : R^3 → R^2. If v =
, then
FA(v) = Av =
So, the image of vector v =
is vector Av =
This function is obviously linear:
FA(u + v) = A(u + v) = Au + Av = FA(u) + FA(v), FA(ku) = A(ku) = k · Au = kFA(u).
