- #1

TheFallen018

- 52

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I'm trying to get my head around linear transformations, and there are a few things I'm not grasping too well. I'm trying to understand combinations of linear transformations, but I can't find a lot of clear information on them. As far as I can tell, any two linear transformations of the same dimensions should be able to be combined into a single transformation. Is there a clear and easy way to prove this though?

Here's an example of what I mean.

Let F and G be $\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ transformations. Define a function $H :$ $\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ by $H(u) = 2F(u)-G(u)$. Show that $H$ is also a linear transformation.

I was thinking of trying to show it using standard matrices of linear transformations, but I'm not sure if that's the best approach.

If you could point me in the right direction, that would be really helpful. Thanks heaps! :)