Help with a basic linear algebra proof

In summary, the task is to prove that if the magnitude of the vector sum of two nonzero vectors x and y in Rn is equal to the sum of their individual magnitudes, then y must be equal to cx for some c > 0. The first part of the proof has been completed, and now the focus is on proving the second part. The equation ||x + y||^2 = (x + y) . (x + y) and the properties of vectors may be useful in this task.
  • #1
paulrb
18
1

Homework Statement



Let x and y be nonzero vectors in Rn. Prove ||x+y|| = ||x|| + ||y|| if and only if y = cx for some c > 0.

Homework Equations



Formula for vector magnitude, basic properties of vectors, possibly other vector formulas

The Attempt at a Solution



I have proved the first part, that is, that ||x+y|| = ||x|| + ||y|| if y=cx for some c > 0.

Now I have to prove the second part:
If ||x+y|| = ||x|| + ||y|| there is some c > 0 such that y = cx.

I don't know how to begin proving this. I'm guessing I need to find c using the equation. However I can't think of a method that allows me to do this.

If someone could give me hint or help me get started, I would greatly appreciate it. Thanks.
 
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  • #3
Thank you, that was enough for me to figure it out :)
 

FAQ: Help with a basic linear algebra proof

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of vector spaces, linear transformations, and systems of linear equations. It involves the use of algebraic methods to solve problems related to these concepts.

What is a proof in linear algebra?

A proof in linear algebra is a logical argument that shows the validity of a statement or theorem using mathematical principles and definitions. It involves the use of mathematical symbols and notation to demonstrate the truth of a given statement.

How do I approach a basic linear algebra proof?

When approaching a basic linear algebra proof, it is important to first understand the definitions and concepts involved. Then, carefully read and analyze the given statement and break it down into smaller parts that can be proven. Use known theorems and properties to support your argument and make sure to provide clear and concise explanations.

What are some common techniques used in linear algebra proofs?

Some common techniques used in linear algebra proofs include direct proof, proof by contradiction, proof by induction, and proof by contrapositive. These methods involve using logical reasoning and mathematical operations to demonstrate the truth of a statement.

What are some tips for writing a successful linear algebra proof?

Some tips for writing a successful linear algebra proof include understanding the definitions and concepts involved, breaking down the statement into smaller parts, using clear and concise language, and providing logical and well-supported arguments. It is also important to check for errors and make sure all steps are justified and valid.

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