- #1
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
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.
micromass said:Since when is [itex]3^a + 3^b = 3^{a+b}[/itex]?
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.
To determine if a transformation is linear, you can use the following criteria:
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.
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.
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.