# Combination of Linear Transformations

TheFallen018
Hello,

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! :)

Gold Member
MHB
Hello,

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! :)
The definition of a linear transformation is that it is a transformation that preserves addition and scalar multiplication.

So if $F : \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ is a linear transformation, that means that $F(u+v) = F(u) + F(v)$ and $F(\lambda u) = \lambda F(u)$ (for all $u,v\in\Bbb{R^n}$ and $\lambda\in\Bbb{R}$).

You have to check that if $F$ and $G$ have those properties then so does $H$.

HOI
Hello,

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.
? Doesn't the problem say that F and G are linear transformations? If not then this is simply not true!

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! :)