Cauchy Schwarz Proof Question

In summary, the task is to prove that |x'y| <= ||x|| ||y|| for vectors x and y, using the given information that ||x|| is the norm of x and x' is the transpose of x. The correct solution involves placing the absolute value signs around x'y rather than ||x|| and ||y||, resulting in the conclusion that |x'y| <= ||x|| ||y||.
  • #1
DiscipulusHum
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Homework Statement


Prove that |x'y| <= ||x|| ||y|| for vectors x, y


Homework Equations


||x|| is the norm of x
x' is the transpose of x

The Attempt at a Solution


||(x/||x||)-(y/||y||)|| = [ (x'/||x|| - y'/||y||)(x/||x|| - y/||y||) ]^1/2 = [-1/(||x||*||y||) (x'y + y'x) +2]^1/2
we know that this norm is positive or zero, so:
[-1/(||x||*||y||) (x'y + y'x) +2]^1/2 > 0
-> x'y + y'x < 2||x||||y||
x'y = y'x because they are scalars and the transpose of a scalar doesn't change its value
-> 2x'y < 2||x||||y||
-> x'y < ||x||||y||

My result doesn't have absolute values around x'y; what am I doing wrong?
 
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  • #2




It looks like you are on the right track, but you have made a small mistake. The absolute value signs should actually be around x'y, not ||x|| and ||y||. This is because the norm of a vector is always positive, so there is no need for the absolute value signs around ||x|| and ||y||. The correct solution would be:
|x'y| = |y'x| <= ||y|| ||x|| = ||x|| ||y||
Therefore, |x'y| <= ||x|| ||y||.

I hope this helps clarify things for you! Keep up the good work.
 

Question 1: What is the Cauchy Schwarz Proof?

The Cauchy Schwarz Proof, also known as the Cauchy-Schwarz inequality, is a mathematical theorem that states that the inner product of two vectors is always less than or equal to the product of their norms. It is used to prove various mathematical and scientific concepts, and has applications in fields such as geometry, statistics, and physics.

Question 2: How is the Cauchy Schwarz Proof used in real life applications?

The Cauchy Schwarz Proof has numerous applications in various fields. In physics, it is used to prove the uncertainty principle in quantum mechanics. In statistics, it is used to establish the correlation coefficient between two variables. It is also used in optimization problems and in proving geometric theorems.

Question 3: What is the history behind the Cauchy Schwarz Proof?

The Cauchy Schwarz Proof is named after mathematicians Augustin-Louis Cauchy and Hermann Amandus Schwarz, who independently discovered the inequality in the 19th century. However, its origins can be traced back to the works of mathematicians such as Leonhard Euler and Joseph-Louis Lagrange in the 18th century.

Question 4: What are some key concepts related to the Cauchy Schwarz Proof?

Some key concepts related to the Cauchy Schwarz Proof include inner product spaces, vector norms, and orthogonality. Understanding these concepts is essential in applying the Cauchy Schwarz Proof in various mathematical and scientific problems.

Question 5: How can I learn more about the Cauchy Schwarz Proof?

There are many resources available to learn more about the Cauchy Schwarz Proof. You can refer to textbooks on linear algebra, calculus, or geometry to understand the concepts and applications of the inequality. There are also online tutorials, videos, and lectures available for further explanation and practice problems.

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