What is the Relationship Between Injectivity and Surjectivity in Linear Algebra?

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In summary, The conversation is about a proof involving real numbers and a function that maps from real^2 to real^2. The goal is to prove that the function is both injective and surjective. The person is having trouble understanding the proof and asks for help. They mention the rank-nullity theorem and a hint from their book, but are still confused.
  • #1
kylem
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I'm having a lot of trouble figuring out how to do this proof:

Given real numbers a, b, c, d, let f: real^2 ----> real^2 be defined by f(x, y) = (ax + by, cx + dy). Prove that f is injective if and only if f is surjective.

If anybody could help, that would be great.
 
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  • #2
Do you know what the rank-nullity theorem is?
 
  • #3
Office_Shredder said:
Do you know what the rank-nullity theorem is?

A quick google search reveals that it appears to be dealing with some linear algebra I don't quite understand...

My book gives me the following hint for this proof: Consider two cases, depending on whether ad - bc = 0.

Not sure if that helps, I'm pretty confused here.
 

1. What is an injection?

An injection is a function between two sets that maps each element in the first set to a unique element in the second set. In other words, no two elements in the first set can map to the same element in the second set.

2. How is an injection different from a surjection?

An injection is a one-to-one function, meaning each element in the first set has a unique mapping to an element in the second set. A surjection, on the other hand, is an onto function where every element in the second set has at least one element in the first set that maps to it.

3. What is the purpose of injections and surjections in mathematics?

Injections and surjections are used to describe the relationship between two sets and their elements. They are important in various mathematical concepts such as functions, relations, and set theory.

4. How can you determine if a function is an injection or a surjection?

To determine if a function is an injection, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not an injection. To determine if a function is a surjection, you can check if every element in the range of the function has at least one corresponding element in the domain.

5. Can a function be both an injection and a surjection?

Yes, a function can be both an injection and a surjection. This type of function is called a bijection, where each element in the first set has a unique mapping to an element in the second set, and every element in the second set has at least one corresponding element in the first set.

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