Combination of Linear Transformations

In summary, the author is trying to understand combinations of linear transformations, but doesn't understand it well. If you could point the author in the right direction, that would be very helpful.
  • #1
TheFallen018
52
0
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! :)
 
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  • #2
TheFallen018 said:
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 $H$.
 
  • #3
TheFallen018 said:
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! :)
 

What is a combination of linear transformations?

A combination of linear transformations is a mathematical operation where multiple linear transformations are applied to a vector in a specific order. This results in a new vector that is the product of all the individual transformations.

How is a combination of linear transformations represented?

A combination of linear transformations is typically represented as a single matrix, where each column represents an individual transformation. The transformations are applied from right to left, with the last transformation being applied first.

What is the difference between a combination of linear transformations and composition of functions?

A combination of linear transformations and composition of functions are similar concepts, but they operate on different types of mathematical objects. A combination of linear transformations operates on vectors, while composition of functions operates on functions. Additionally, a combination of linear transformations is represented as a matrix, while composition of functions is represented as a function.

Can a combination of linear transformations be reversed?

Yes, a combination of linear transformations can be reversed by finding the inverse of the matrix representing the combination. This will result in a new matrix that, when applied to the combined vector, will result in the original vector.

What are some real-world applications of combinations of linear transformations?

Combinations of linear transformations are used in various fields, such as computer graphics, robotics, and physics. They are used to transform and manipulate objects in 3D space, simulate physical systems, and control the movements of robots. They are also used in data compression techniques, such as JPEG image compression.

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