Thread: Proving that the image is a subspace

1. Proving that the image is a subspace

If V and W are two vector spaces over the same field K, and T : V→W is a linear mapping, how do I prove that the image of the set T, given by
Im(T) = (w belongs to W : there exists v belongs to V such that T(v)= w),
Is a subspace of W

2. Originally Posted by MuhTheKuh
If V and W are two vector spaces over the same field K, and T : V→W is a linear mapping, how do I prove that the image of the set T, given by
Im(T) = (w [IMG]file:///C:/Users/Tristan/AppData/Local/Temp/moz-screenshot-1.png[/IMG]belongs to [IMG]file:///C:/Users/Tristan/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG][IMG]file:///C:/Users/Tristan/AppData/Local/Temp/moz-screenshot.png[/IMG] W : there exists v belongs to[IMG]file:///C:/Users/Tristan/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG] V such that T(v)= w),
Is a subspace of W

as you can see, your images are not showing up...

3. Thanks. I was just trying to get the "belongs to" sign in there, but when that did not work out I just left it and completely forgot about that one.
Sorry.

4. Originally Posted by MuhTheKuh
Thanks. I was just trying to get the "belongs to" sign in there, but when that did not work out I just left it and completely forgot about that one.
Sorry.
So you're trying to prove that if $V,W$ are two $K$-spaces and $T:V\to W$ is a linear transformation then $T\left(V\right)$ is a subspace of $W$, right? But this amounts to proving that $x,y\in T\left(V\right)\text{ and }\alpha,\beta\in K\implies \alpha x+\beta y\in T\left(V\right)$. But, by definition $x=T(x')$ and $y=T(y')$ for some $x',y'\in V$ so that you're trying to prove that $\alpha T(x')+\beta T(y')\in T\left(V\right)$ but $\alpha T(x')+\beta T(y')=\cdots$?

5. aT(x')+bT(y')= ax+by

but is that enough to prove it?

6. Originally Posted by MuhTheKuh
aT(x')+bT(y')= ax+by

but is that enough to prove it?
No. Try using the linearity of $T$