# Definition of Image of a linear transformation

1. Feb 3, 2016

### says

1. The problem statement, all variables and given/known data
The image of a linear transformation = columnspace of the matrix associated to the linear transformation.
More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X that are mapped to Y. This may or may not be equal to the all of the vectors in subspace X and subspace Y.

I was going to say, the Im(T) = all of the vectors in X that are mapped to Y, but the definition sounds a bit 'muddier', but I'm not entirely sure. Hence my post.

I usually draw a picture like this: http://thejuniverse.org/PUBLIC/LinearAlgebra/MATH-232/Unit.8/Presentation.1/Section7B/image.png to go with my definition, but I wanted to check.

I saw a reference book that said 'the image of a linear transformation f : V → W is the set of vectors the linear transformation maps to.'

2. Relevant equations

3. The attempt at a solution

2. Feb 3, 2016

### blue_leaf77

The image of a linear transformation $T$ defined by $T:V\rightarrow W$ is the set of all vectors which are the result of the linear map $T$ applied on any vector in $V$. Which means $\textrm{Im}(T)$ is a subset of $W$, not of $V$. In this sense, the last definition:
is the correct one.

3. Feb 3, 2016

### HallsofIvy

Staff Emeritus
No! The "image of a linear transformation" is set of vectors in Y such that some vector in X is mapped to it. Taking the linear transformation to be "T", the image is "all y in Y such that there exist x in X such that Tx= y."

The image is a subset (actually subspace) of Y, not X. It may or may not be all of Y.

Again, no, no, no! Im(T) = all vectors in Y that vectors in X are mapped to it.

The pink area in your picture is the image.

Yes, that is correct, NOT what you said!