Determine if a transformation is linear.

In summary, the conversation is discussing two separate transformations and whether they are linear. The individual responding has attempted to prove if T(u+v) = T(u)+T(v) and T(cu)=cT(u) for both transformations, but concludes that neither satisfy the definitions of linearity. The original poster then questions the validity of one of the equations and the responder clarifies their point. The conversation ends with the responder pointing out that simply writing things down does not indicate whether they are believed to be correct or incorrect.
  • #1
thatguythere
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Homework Statement


Please see attached files and let me know if I am correct or not.

Homework Equations





The Attempt at a Solution

 

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  • #2
thatguythere said:

Homework Statement


Please see attached files and let me know if I am correct or not.

Homework Equations





The Attempt at a Solution


Can't say whether you are correct or not. You just wrote a bunch of stuff down. Are they linear or not?
 
  • #3
What do you mean, I just wrote a bunch of stuff down? They are two separate transformations. I applied arbitrary vectors to them and attempted to prove if T(u+v) = T(u)+T(v) as well as T(cu)=cT(u)
In the first transformation, it appears that the first definition is not satisfied and in the second problem, the second definition is not satisfied. Therefore, I do not believe either are linear, however I am not certain if I am doing this properly.
 
  • #4
thatguythere said:
What do you mean, I just wrote a bunch of stuff down? They are two separate transformations. I applied arbitrary vectors to them and attempted to prove if T(u+v) = T(u)+T(v) as well as T(cu)=cT(u)
In the first transformation, it appears that the first definition is not satisfied and in the second problem, the second definition is not satisfied. Therefore, I do not believe either are linear, however I am not certain if I am doing this properly.

That's exactly what I was asking for. I wanted to know your conclusions from what you wrote down. Correct that thay are both not linear. If you want to prove that just come up with specific examples of u, v and c (with numbers in them) where T(u+v)=T(u)+T(v) or T(cu)=cT(u) don't work.
 
  • #5
Since when is [itex]3^a + 3^b = 3^{a+b}[/itex]?
 
  • #6
micromass said:
Since when is [itex]3^a + 3^b = 3^{a+b}[/itex]?

That was my point. Just writing these things down doesn't say whether the poster believes them to be correct or incorrect. I'm not sure what writing a check next to it means.
 

Related to Determine if a transformation is linear.

1. What is a linear transformation?

A linear transformation is a mathematical function that maps a set of input values to a set of output values, while preserving the properties of linearity. This means that for any two points in the input space, the resulting output points will lie on a straight line.

2. How can I determine if a transformation is linear?

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

  • The transformation must preserve addition, meaning that if you add two points in the input space, the resulting output points will also be added.
  • The transformation must preserve scalar multiplication, meaning that if you multiply a point in the input space by a constant, the resulting output point will also be multiplied by that constant.
  • The transformation must preserve the zero vector, meaning that the transformation of the zero vector in the input space must be the zero vector in the output space.

3. What is the importance of linearity in transformations?

Linearity is important in transformations because it allows us to manipulate and analyze data using simpler mathematical operations. Linear transformations also have many useful properties that make them easier to work with, such as being able to easily invert and compose with other transformations.

4. Can a transformation be both linear and non-linear?

No, a transformation can only be either linear or non-linear. If a transformation does not satisfy the criteria for linearity, then it is considered non-linear.

5. Are all linear transformations one-to-one?

No, not all linear transformations are one-to-one. A linear transformation can be one-to-one (injective), onto (surjective), or both (bijective) depending on the properties of the transformation. For example, a transformation that scales all points in the input space by a constant factor is not one-to-one, as multiple input points may map to the same output point.

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