Exploring Orthogonality in Linear Algebra: A Simple Proof of a Common Identity

In summary, orthogonality is a concept in mathematics and science that refers to the relationship between two objects that are perpendicular to each other. It is important because it allows for simplification and solving of complex problems, and is used in fields such as physics, engineering, and computer science. Orthogonality differs from parallelism, which refers to objects that are equidistant and never intersect. Orthogonal vectors can be both dependent and independent, and in data analysis and machine learning, they are used to reduce the dimensionality of data by eliminating redundant information.
  • #1
EvLer
458
0
In one of my homework problems we are asked to prove this:

|X + Y|^2 + |X - Y|^2 = 2|X|^2 + 2|Y|^2

It seems quite simple: I just expanded both of them (cancelled 2(X dot Y) terms) and came up with the expression on the right. Is there something I am missing or is it really that simple?

Thanks !
 
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  • #2
It really is!
 
  • #3
linear algebra makes many questions in geometry rather trivial. for this reason high schools should teach the subject, instead of calculus, where they usually do an extremely bad job.
 

1. What is orthogonality?

Orthogonality refers to the relationship between two vectors or functions that are perpendicular to each other. In other words, they have a zero inner product, meaning they do not share any common components.

2. Why is orthogonality important in mathematics and science?

Orthogonality is important because it allows us to simplify and solve complex problems. It is a fundamental concept in linear algebra and is used in fields such as physics, engineering, and computer science.

3. How is orthogonality different from parallelism?

While both orthogonality and parallelism involve the relationship between two objects, they are not the same. Orthogonality refers to perpendicularity, while parallelism refers to objects that are equidistant and never intersect.

4. Can orthogonal vectors be dependent or independent?

Orthogonal vectors can be both dependent and independent. If two vectors are orthogonal, they are always independent. However, if multiple vectors are orthogonal to each other, they can be dependent on each other.

5. How is orthogonality used in data analysis and machine learning?

In data analysis and machine learning, orthogonality is used to reduce the dimensionality of data. By finding orthogonal vectors, we can eliminate redundant information and simplify the data, making it easier to analyze and process.

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