Having difficulty understanding what the Range of a linear transformation is.

[phantomcow2]

One of the topics in my linear algebra course is kernel and range of a linear transformation. I have a firm understanding of what the kernel is: the set of vectors such that it maps all inputs to the zero vector. Range, however, remains nebulous to me. My textbook says that the range is "THe set of all vectors in W that are images under T of at least one vector in V."

I'm not sure what it means to be "an image under T." Could somebody explain this to me? I'd just like to have this concept clarified. Thanks.

 

[Newtime]

One of the topics in my linear algebra course is kernel and range of a linear transformation. I have a firm understanding of what the kernel is: the set of vectors such that it maps all inputs to the zero vector. Range, however, remains nebulous to me. My textbook says that the range is "THe set of all vectors in W that are images under T of at least one vector in V."

I'm not sure what it means to be "an image under T." Could somebody explain this to me? I'd just like to have this concept clarified. Thanks.

See the attachment.

 

[phantomcow2]

Wow, that is a wordy explanation and exactly what I needed. Thanks :).

 

[radou]

The range and image are the same thing. But the codomain isn't. If a mapping is surjective, then they're all the same set, but in general, not.

 

[blue_raver22]

The range of a linear transformation refers to the set of all output vectors that can be obtained by applying the transformation to at least one input vector. In other words, it is the set of all possible values that the transformation can produce.

To better understand this concept, let's consider an example. Imagine we have a linear transformation that takes a 2D vector and rotates it by 90 degrees counterclockwise. The range of this transformation would be all possible 2D vectors that can be obtained by rotating any input vector by 90 degrees counterclockwise. So, the range would include vectors such as (1,0), (0,1), (-1,0), (0,-1) and all their combinations.

The term "image under T" refers to the output vector that is obtained after applying the transformation T to a specific input vector. In our example, the image under T of the vector (1,0) would be (-1,0) since that is the result of rotating (1,0) by 90 degrees counterclockwise.

I hope this explanation clarifies the concept of range for you. It is an important concept in linear algebra as it helps us understand the full scope of a linear transformation and its capabilities.