Discovering the Image of a Linear Map: Methods and Examples

[Marin]

Hi all!

Does anyone know a general method for determining the image of a lin map?

I´m aware of the definition if im, but how could I determine it. Maybe it would be useful to show this on some examples :)

 

[HallsofIvy]

What exactly do you mean by "determining the image". Apply the linear map to each of the basis vectors of a the domain space gives you a set that spans the image. You can reduce that to get a basis for the image.

If you are looking at a matrix, you can "column reduce" the matrix and and the columns of the reduced matrix are a basis for the image. If you have only learned "row reduction", swap rows for columns (the "transpose") and row reduce. The rows of the reduced matrix form a basis for the image.

 

[Hurkyl]

Most computational linear algebra questions are best approached by first formulating the question in terms of matrices.

e.g. HallsofIvy suggests to capture the notion of image somehow via the column space of an appropriate matrix.

 

[mathwonk]

a standard technique for proving the image is the whole codomain is to show the rank of the map equals the dimension of the codomain. e.g. if the domain and codomain have the same dimension, then it suffices to show the map is injective.

 

[Marin]

thanks to all of you!

I think it`s all getting somehow clearer to me :)

(sorry for the inexactly asked question, I started my linear algebra course about a month ago and I`m still getting used to the level of abstraction it requires)

 

[mathwonk]

In general it is not trivial to determine the image of a linear map especially in infinite dimensions. e.g. the main theorem of ordinary differential equations says certain linear differential operators acting on smooth functions, have as image the space of all smooth functions.