Proof of Cauchy's Inequality .... Sohrab Proposition 2.1.23

In summary, the student is trying to figure out how to solve an equation that has real numbers as its inputs, and they are not sure how to use the fact that the discriminant is less than or equal to zero.
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Homework Statement



I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with Proposition 2.1.23/Exercise 2.1.24 (Exercise 2.1.24 asks readers to prove Proposition 2.1.23)

Proposition 2.1.23/Exercise 2.1.24 reads as follows:

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In the above text by Sohrab, we read the following:

" ... ... Observe that for any ##t \in \mathbb{R}, F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0## and look at the discriminant ##Z^2 - XY## of ##F(t)##. ... ... "Can someone please explain how one proceeds after determining the discriminant and what principles/properties of the discriminant are used ...

Homework Equations



The relevant information is given in the problem statement ... basic properties of the real numbers are assumed ...

The Attempt at a Solution



my working so far on the exercise is as follows:

##F(t) = \sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0##

Now ... ... ##\sum_{ i = 1 }^n ( x_i - t y_i )^2 \ge 0##

##\Longleftrightarrow \sum_{ i = 1 }^n ( x_i^2 - 2t x_i y_i + t^2 y_i^2) \ge 0##

##\Longleftrightarrow \sum_{ i = 1 }^n x_i^2 - 2 t \sum_{ i = 1 }^n x_i y_i + t^2 \sum_{ i = 1 }^n y_i^2 \ge 0## ... ... ... (1)... now let ##X = \sum_{ i = 1 }^n x_i^2##, ## \ ## ##Y = \sum_{ i = 1 }^n y_i^2 \ ## and ## \ Z = \sum_{ i = 1 }^n x_i y_i##Then (1) becomes ##X - 2t Z + Y t^2 \ge 0##

and the discriminant, ##D = 4Z^2 - 4XY##BUT ... how do we proceed from here .. ... and exactly what properties of the discriminant do we use ...Help will be appreciated,

Peter
 

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If [itex]ax^2 + bx + c \geq 0[/itex] for all [itex]x \in \mathbb{R}[/itex], then its roots are either both zero or a complex conjugate pair.
 
  • #3
pasmith said:
If [itex]ax^2 + bx + c \geq 0[/itex] for all [itex]x \in \mathbb{R}[/itex], then its roots are either both zero or a complex conjugate pair.

Thanks pasmith ... BUT ... I am not sure how to use that fact ...

PeterEDIT

On further reflection, then the discriminant is less than or equal to zero ... so the result follows ...

I think that that is correct ...

Thanks ...
 
Last edited:

Related to Proof of Cauchy's Inequality .... Sohrab Proposition 2.1.23

1. What is Cauchy's Inequality?

Cauchy's Inequality is a mathematical concept named after the French mathematician Augustin-Louis Cauchy. It states that for any two sequences of real numbers, the sum of the products of corresponding terms is less than or equal to the product of the sums of the sequences. In other words, it states that the dot product of two vectors is always less than or equal to the product of their magnitudes.

2. What is Sohrab Proposition 2.1.23?

Sohrab Proposition 2.1.23 is a mathematical proposition that is derived from Cauchy's Inequality. It states that the absolute value of the dot product of two vectors is always less than or equal to the product of their magnitudes. This is a special case of Cauchy's Inequality and is often used in mathematical proofs and calculations.

3. What is the significance of Cauchy's Inequality in mathematics?

Cauchy's Inequality is a fundamental concept in mathematics and has applications in various fields such as linear algebra, calculus, and statistics. It is often used in mathematical proofs and is a crucial tool in establishing theorems and solving problems in these fields. It also has practical applications in physics, engineering, and economics.

4. Can Cauchy's Inequality be applied to vectors in any dimension?

Yes, Cauchy's Inequality can be applied to vectors in any dimension. However, it is most commonly used in two or three dimensions, as it is easier to visualize and understand in these cases. In higher dimensions, the concept still holds true, but it becomes more challenging to visualize and prove mathematically.

5. How is Cauchy's Inequality related to other mathematical concepts?

Cauchy's Inequality is closely related to other mathematical concepts, such as the Cauchy-Schwarz Inequality, the Triangle Inequality, and the Hölder's Inequality. These concepts all involve the comparison of the dot product of two vectors to the product of their magnitudes and have various applications in mathematics and other fields. Cauchy's Inequality is also related to the concept of orthogonality, which is used in linear algebra and geometry.

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