Lindemann–Weierstrass theorem?

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In summary, the Lindemann-Weierstrass theorem states that an increasing sequence of real numbers can be used to create a set of exponential functions that are linearly independent. This can be proven by showing that any set of coefficients that results in a sum of zero for all values of t must have all coefficients equal to zero. Additionally, the set {1, sqrt(2), sqrt(3), sqrt(6)} is also linearly independent over the rational numbers, as proven by the fact that the irrational numbers \sqrt{2} and \sqrt{3} cannot be expressed as rational numbers. This can be proven by induction and by letting t approach infinity in the equation a_1e^{x_1t}+
  • #1
phantomprime
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Lindemann–Weierstrass theorem??

Let X1, X2, X3,...Xn be an increasing sequence of real numbers. Prove that the n exponential functions e^x1t,e^x2t...e^xnt are linearly independent.

My question is how? I look at examples but how do I go about explaining this?

______________________________________...

Show that the set {1, sqrt(2), sqrt(3), sqrt(6)} is linearly independent over the rational numbers
 
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  • #2


(sorry, ignore this post)
 
  • #3


I think if you just keep picking values of t, you will get a system of linear equations that is overdetermined (and unsolvable since the columns of rows of constants in the matrix are linearly independent).
 
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  • #4


Both of those look pretty straight forward from the definition of "independent".

Given [itex]a_1e^{x_1t}+ a_2e^{x_2t}+ \cdot\cdot\cdot+ a_ne^{x_nt}= 0[/itex]
for all t, prove that [itex]a_1= a_2= \cdot\cdot\cdot= a_n= 0[/itex]
I suspect it would be simplest to prove this by induction on n.

For the second one, you want to prove that if [itex]a+ b\sqrt{2}+ c\sqrt{3}+ d\sqrt{6}= 0[/itex], and a, b, c are rational numbers, then a= b= c= d= 0. You will, of course, use the fact that [itex]\sqrt{2}[/itex] and [itex]\sqrt{3}[/itex] are not rational.
 
  • #5


How would one go from the kth case to the k+1th?
 
  • #6


If we multiply both sides of

[tex]a_1e^{x_1t}+ a_2e^{x_2t}+ \cdots + a_ne^{x_nt}= 0[/tex]

by [itex]e^{-x_n t}[/itex], we get

[tex]a_1e^{(x_1 - x_n)t}+ a_2e^{(x_2 - x_n)t}+ \cdots + a_ne^{(x_{n-1} - x_n)t} + a_n = 0.[/tex]

Now let [itex]t \to \infty[/itex].
 

1. What is the Lindemann-Weierstrass theorem?

The Lindemann-Weierstrass theorem is a mathematical theorem that states that for any non-zero algebraic numbers α1, ..., αn, and any real numbers x1, ..., xn, the product α1x1⋯αnxn is a transcendental number, meaning it is not a root of any non-zero polynomial with integer coefficients.

2. Who discovered the Lindemann-Weierstrass theorem?

The Lindemann-Weierstrass theorem was first proved by German mathematician Ferdinand von Lindemann in 1882, building on the previous work of German mathematician Karl Weierstrass.

3. What is the significance of the Lindemann-Weierstrass theorem?

The Lindemann-Weierstrass theorem has significant implications in the field of transcendental numbers, which are numbers that are not algebraic. It provides a method for proving the transcendence of certain numbers, including the famous mathematical constant π. It also has applications in other areas of mathematics, such as complex analysis and algebraic geometry.

4. Can the Lindemann-Weierstrass theorem be extended to higher dimensions?

Yes, there is a multidimensional version of the Lindemann-Weierstrass theorem, known as the Gelfond-Schneider theorem. This theorem states that for any algebraic numbers α and β, where α is non-zero and β is irrational, the number αβ is transcendental. This theorem can be extended to higher dimensions by considering multiple algebraic numbers and irrational numbers.

5. What are some applications of the Lindemann-Weierstrass theorem?

The Lindemann-Weierstrass theorem has applications in various fields, such as number theory, complex analysis, and algebraic geometry. It has been used to prove the transcendence of numbers such as e, the Euler-Mascheroni constant, and certain values of trigonometric functions. It also has applications in cryptography, where the transcendence of certain numbers is important for ensuring the security of encryption algorithms.

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