# Example of a linear transformation T: p4 -> R^4 thats onto

• sportlover36
In summary, the conversation discusses finding an example of a linear transformation from p4 to R^4 that is onto, and providing advice on how to prove that the transformation is onto. The suggestion is to first give a simple example of a linear transformation, and then use the definition of "onto" to prove it. The conversation ends with the individual still feeling confused and in need of further guidance.
sportlover36
in my book it asks me to give an example of a linear transformation T: p4 -> R^4 that's onto. I have to prove that T is onto and a linear transformation...can someone give me some advice?

Actually, start with giving a simple example of a linear transformation- you have to have the transformation before you can prove it is one-to-one!

Then use the definition of "onto".

im still confused

Can you think of any transformation that takes a polynomial such as x^4 + 5x^3 - x^2 + 2x - 5 to a vector with four components?

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the operations of vector addition and scalar multiplication. In other words, it takes in a vector and outputs another vector that is still within the same vector space.

## 2. What does it mean for a linear transformation to be onto?

A linear transformation is onto if every element in the output space is mapped to by at least one element in the input space. In other words, every vector in the output space has a pre-image in the input space.

## 3. Can you provide an example of a linear transformation that is onto?

Yes, the transformation T: R^3 -> R^2 defined by T(x, y, z) = (x + y, 2y + z) is onto. This means that for any vector (a, b) in R^2, there exists a vector (x, y, z) in R^3 such that T(x, y, z) = (a, b).

## 4. How do you determine if a linear transformation is onto?

To determine if a linear transformation is onto, you can use the rank-nullity theorem, which states that the rank of the transformation plus the dimension of the null space must equal the dimension of the input space. If the rank of the transformation is equal to the dimension of the output space, then the transformation is onto.

## 5. Why is it important for a linear transformation to be onto?

A linear transformation being onto means that every element in the output space has a pre-image in the input space, which is important for applications in mathematics, physics, and engineering. It allows for the manipulation and transformation of data in a way that preserves important properties and relationships, making it a useful tool in problem solving and analysis.

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