Introduction to LATEX
Exercise 6
1. Define four new commands that typeset R, C, N and Z.
2. Define a command \dot{...}{...} which accepts two arguments and
which produces the dot product of its arguments in math mode. So
\dot{a}{b} would produce e.g. ha, bi. You may also use the notation
(a | b) or ha | bi.
3. Define a numbered environment for theorems and use it to typeset the
following theorems: (use your own command for natural numbers)
Theorem 1. If k ∈ N and k ≥ 2 then k can be written as a product of
prime numbers.
Theorem 2. If the prime number p divides the product ab then p divides either a or b.
4. Define a numbered environment for definitions using the style definition
and use it to typeset the following: (use your own command for integers)
Definition 1. Let a, b ∈ Z, a 6= 0. If there exists k ∈ Z such that
b = ka, we say that the number a divides the number b.
5. Define an unnumbered environment for remarks using the style remark
and use it to typeset:
Remark. The equation a = qb + r, where 0 ≤ r < |b|, is called the
division algorithm. Moreover, a is called the dividend, b the divisor, q
the quotient and r the remainder.
6. Define a numbered environment for lemmas so that lemmas are numbered using the same counter as theorems. Use the environment and
command \dot{}{} to typeset the following: (use your own command
for real numbers)
Lemma 3. If x, y ∈ Rn then
|hx, yi| ≤ |x| · |y|.
7. Label the Lemma 3 above and refer to it by typesetting: (use your own
command for complex numbers)
Equality holds in Lemma 3 if and only if x = λy for some λ ∈ C.
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Optional exercises:
8. Typeset the following theorem using your own theorem environment
and the proof with proof environment: (Tip: \ker . The two parts of
the proof may be separated with a paragraph change which may look
different than the one below.)
Theorem 4. Linear mapping A : V → W is injective if and only if
ker A = {0}.
Proof. Let ker A = {0}. If Ax = Ay, then A(x − y) = Ax − Ay = 0.
So x − y ∈ ker A = {0}. Hence x = y and A is injective.
Let us assume conversely, that A : V → W is injective and x ∈ ker A.
Then Ax = 0 = A0. Injectivity implies that x = 0 or ker A = {0}.
9. Define a new command \dx which produces the symbol dx inside
integral sign with correct spacing. Define the command so that the default variable is x, which can be changed at will. The command works
as follows:
\int_A f(x) \dx = \int_A f(y) \dx[y]
Z
Z
f (y) dy
f (x) dx =
A
A
10. Experiment on how to control the numbering of your own theorem
enviroments by defining them with an optional counter argument.
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