# Verify Linear Transformations: 3.4a, 3.4b

• laminatedevildoll
In summary, the conversation is about verifying the correctness of problems in two attachments, 3.4b and 3.4a. The first attachment includes questions about a function T and its kernel and image, while the second attachment involves proving that T is one-to-one and onto. The person seeking help is unsure about their proofs and would appreciate assistance. However, they later understand how to prove T is onto.
laminatedevildoll
I've uploaded a document which I am currently working on. I would like to verify if I am doing these problems correctly. Thank you.

In the first attachment (3.4b)
For 1.
a. 4x^3-2x
b. T(P)=0
ker T={C:C $$\in$$R}
Im T = {P|P is less than degree 3 or less}

c. T is not one to one because P is a constant. T is not onto because it's degree less than 3. I am not sure if I am proving this right. I'd appreciate some help.

For 2.
a. 1/5x^5-1/3x^3+C
b.
ker T={C:C $$\in$$R}
Im T = {P|P is less than degree 5 or less}
T is not one to one because C=0. But, T is on-to. How do I prove this?

In the second attachment (3.4a) part 4.
I proved that T is one to one because T(f)=T(g), f=g How do I prove that this is on-to?

#### Attachments

• 3-4b.pdf
54.4 KB · Views: 247
• 3-4a.pdf
69.2 KB · Views: 229
nvm, I understand it.

Overall, your work looks good and your reasoning is correct. Here are some suggestions for further clarifications:

For 3.4b, 1:
a. Your answer for T(P) is correct, but you could also write it as T(P) = 0x^3 + 0x^2 + 0x - 0. This might make it clearer that T(P) is a polynomial of degree less than or equal to 3.
b. Your answer for ker T and Im T are correct, but you could also add a brief explanation for why they are the same.
c. Your reasoning is correct, but you could also mention that a constant polynomial cannot be mapped to a non-constant polynomial, so T is not onto.

For 3.4b, 2:
a. Your answer for T(P) is correct, but you could also write it as T(P) = 1/5x^5 + (-1/3)x^3 + C. This might make it clearer that T(P) is a polynomial of degree less than or equal to 5.
b. Again, your answers for ker T and Im T are correct, but you could add a brief explanation for why they are the same.
c. To prove that T is onto, you could show that for any polynomial of degree less than or equal to 5, there exists a polynomial P such that T(P) = that polynomial. In other words, show that for any polynomial Q(x) of degree less than or equal to 5, you can find a polynomial P such that T(P) = Q(x). This would show that every polynomial of degree less than or equal to 5 is in the image of T, and thus T is onto.

For 3.4a, 4:
Your proof for T being one-to-one is correct. To show that it is onto, you could use a similar approach as in 3.4b, 2c. Show that for any polynomial Q(x), there exists a polynomial P such that T(P) = Q(x). This would show that every polynomial is in the image of T, and thus T is onto.

Overall, your work is correct and well-reasoned. Keep up the good work!

## 1. What is a linear transformation?

A linear transformation is a mathematical concept that describes a function or mapping between two vector spaces that preserves the operations of addition and scalar multiplication. In other words, the output of a linear transformation is a linear combination of its inputs.

## 2. How do you verify if a transformation is linear?

To verify if a transformation is linear, you can use the following criteria:

• Check if the transformation preserves addition: T(u + v) = T(u) + T(v)
• Check if the transformation preserves scalar multiplication: T(ku) = kT(u)
• Verify that the transformation is well-defined for all inputs in the vector space.
If all of these criteria are met, then the transformation is linear.

## 3. What is the difference between 3.4a and 3.4b in Verify Linear Transformations?

In 3.4a, you are verifying if a specific transformation is linear, while in 3.4b, you are verifying if two transformations are equivalent or have the same properties.

## 4. Why is it important to verify linear transformations?

Verifying linear transformations is important because it ensures that the transformation can be used in mathematical calculations and that it accurately represents the relationship between two vector spaces. Additionally, verifying linear transformations is a crucial step in solving problems in fields such as physics, engineering, and computer science.

## 5. Can a linear transformation have different representations?

Yes, a linear transformation can have different representations depending on the choice of basis for the vector spaces involved. This means that the same transformation can be represented by different matrices or equations, but it will still preserve the linear properties.

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