How to check if a transformation is surjective and injective

• caspeerrr
In summary, the Attempt at a Solution is correct, but the methods described are more basic and general than what was provided in the question.
caspeerrr

Homework Statement

I have attached the question. Translated: Suppose T: R^4 -> R^4 is the image so that: ...

Homework Equations

So I did this question and my final answers were correct: 1. not surjective 2. not injective. My method of solving this question is completely different than the answerbook thoug. Is my method correct too?

The Attempt at a Solution

1. If i put the transformation inside a matrix the result would be:
0 1 1 0--------------------------------------0 1 0 1
0 0 3 1 which can be reduced t----0 0 1 -1
0 -1 0 0-------------------------------------- 0 0 0 1
0 -1 0 1--------------------------------------- 0 0 0 0
[/B]
1. learned that a condition for a surjective transformation is that there has to be a pivot position in each row, which is not true: pivot positions are in row 1, 2 and 3, but not 4. So not Surjective.

2. Secondly I learned that a condition for a injective transformation is that there can be no free variables. In the matrix above, there is one column with only zeros. This means that X1 is a free variable, it doesn't matter what value you give to it, it will not affect the final outcome.

Is what i did correct? Thanks in advance.

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You are correct, but the methods you describe are more of cooking recipes than providing understanding for what is actually going on.

In order for a transformation to be surjective, there needs to be at least one ##x## in the domain such that ##f(x) = y## for every ##y## in the codomain.

In order for a transformation to be injective, each ##x## in the domain must be mapped to a unique element ##y## in the codomain.

In the case of linear transformations, it is helpful to think about these concepts in terms of linear independence.

FactChecker
You need to be careful. The concepts of surjective and injective are very basic and general. They are used in situations where pivot elements and matrices are not applicable. So if your methods are different, you may not be learning the basic definitions and methods that you should be learning.

So the question is: How did the book do it and do you understand it? If their method looks more basic and general, you should be using their methods.

My advice would be to look at surjective and injective from a Set Theory perspective. Then see how this definition fits into what you are doing.

FactChecker
These concepts can be tricky to describe. Technically I feel it is important to begin the statement with the element y in the target space, so that the quantifiers are in the right order. One way I like is to say that f is injective if for every y in the target space, there is at most one x in the domain, or source space, such that f(x) = y. Then one can say analogously that f is surjective if for every y in the target space, there is at least one x in the source space such that f(x) = y.

then the OP might check that in case every row has a pivot, that no matter what column vector y he puts on the right, he will be able to find an x such that T(x) = y. Thus T is surjective.

And if every column has a pivot, no matter what column y he puts on the right, he will not be able to find more than one x with T(x) = y. In this last case, injectivity, it will be sufficient to show that if he puts the zero column y on the right, then the only solution is the zero vector x.

1. What is the definition of a surjective transformation?

A surjective transformation, also known as a surjection, is a type of function that maps elements from one set to another, where every element in the range (output) set has at least one element in the domain (input) set that maps to it. In other words, every output value has at least one corresponding input value.

2. How can I check if a transformation is surjective?

To check if a transformation is surjective, you can use the "vertical line test." This involves drawing a vertical line through the graph of the transformation. If the line intersects the graph at more than one point, then the transformation is not surjective. However, if the line intersects the graph at exactly one point, then the transformation is surjective.

3. What is the definition of an injective transformation?

An injective transformation, also known as an injection, is a type of function that maps elements from one set to another, where each element in the range (output) set has at most one element in the domain (input) set that maps to it. In other words, every output value has at most one corresponding input value.

4. How can I check if a transformation is injective?

To check if a transformation is injective, you can use the "horizontal line test." This involves drawing a horizontal line through the graph of the transformation. If the line intersects the graph at more than one point, then the transformation is not injective. However, if the line intersects the graph at exactly one point, then the transformation is injective.

5. Can a transformation be both surjective and injective?

Yes, a transformation can be both surjective and injective. This type of transformation is called a bijective transformation, also known as a bijection. A bijection is a one-to-one correspondence between the input and output sets, meaning that every output value has exactly one corresponding input value, and every input value has exactly one corresponding output value.

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