# Linear Transformations: Proving Rules & Demonstration

• MHB
• Ereisorhet
In summary, the given problem requires demonstrating that the given transformations are linear transformations using the rules of T(u+v) = T(u) + T(v) and T(Lu) = LT(u). For part (a), the summary shows that T(u+v) = T(u) + T(v) by substituting values and comparing. For part (b), T(u+v) = T(u) + T(v) is also shown by substituting values and comparing. Finally, for the last part, the summary shows that T(Lu) = LT(u) by substituting values and comparing.
Ereisorhet
Good afternoon people.
So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if:
T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to develop that and find the demonstration.

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For (a), let $$u= \begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix}$$ and $$u= \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}$$.

We are told that "$$T\begin{pmatrix}x \\ y \\ z \end{pmatrix}= \begin{pmatrix}1 \\ z \end{pmatrix}$$ so $$Tu= \begin{pmatrix}1 \\ u_3\end{pmatrix}$$ and $$Tv= \begin{pmatrix}1 \\ v_3\end{pmatrix}$$. So $$Tu+ Tv= \begin{pmatrix}2 \\ u_3+ v_3 \end{pmatrix}$$.
But $$u+ v= \begin{pmatrix}u_1+ v_1 \\ u_2+ v_2 \\ u_3+ v_3 \end{pmatrix}$$ so $$T(u+ v)= \begin{pmatrix}1 \\ u_3+ v_3\end{pmatrix}$$. Is T(u+ v)= Tu+ Tv?

For (b), let $$u= \begin{pmatrix}u_1 \\ u_2 \end{pmatrix}$$ and $$v= \begin{pmatrix}v_1 \\ v_2 \end{pmatrix}$$. $$Tu= \begin{pmatrix} 2u_1+ u_2 \\ u_1- 3u_2 \\ u_1 \\ u_ 2 \end{pmatrix}$$ and $$Tv= \begin{pmatrix} 2v_1+ v_2 \\ v_1- 3v_2 \\ v_1 \\ v_2 \end{pmatrix}$$. So $$Tu+ Tv= \begin{pmatrix}2u_1+ u_2+ 2v_1+ v_2 \\ u_1- 3u_2+ v_2- 3v_2 \\ u_1+ v_2 \\ u_2+ v_2 \end{pmatrix}$$.
$$u+ v= \begin{pmatrix}u_1+ v_1 \\ u_2+ v_2 \end{pmatrix}$$ so $$T(u+ v)= \begin{pmatrix}2(u_1+ v_1)+ (u_2+ v_2) \\ (u_1+ v_1)- 3(u_2+ v_3) \\ u_1+ v_1 \\ u_2+ v_2 \end{pmatrix}$$.

Do you see that those are the same, so T(u+ v)= Tu+ Tv?

Now we need to show that "T(Lu)= LTu" where L is a "scalar" (a number). $$Lu= \begin{pmatrix}Lu_1 \\ Lu_2 \end{pmatrix}$$ so $$T(Lu)= \begin{pmatrix} 2Lu_1+ Lu_2 \\ Lu_1- 3Lu_2 \\ Lu_1 \\ Lu_2 \end{pmatrix}$$ and $$LTu= L\begin{pmatrix}2u_1+ u_2 \\ u_1- 3u_2 \\ u_1 \\ u_2\end{pmatrix}= \begin{pmatrix}L(2u_1+ u_2) \\ L(u_1- 3u_2) \\ Lu_1 \\ Lu_2 \end{pmatrix}$$. Do you see that T(Lu)= LTu?

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps vectors from one vector space to another, while preserving the basic structure of the original vector space. In other words, it is a function that takes in a vector and outputs another vector, while following certain rules.

## 2. What are the rules for proving a linear transformation?

The rules for proving a linear transformation are:

1. The transformation must preserve vector addition, meaning that the sum of two input vectors must equal the sum of their transformed outputs.
2. The transformation must preserve scalar multiplication, meaning that multiplying a vector by a scalar must equal multiplying its transformed output by the same scalar.

## 3. How do you demonstrate a linear transformation?

To demonstrate a linear transformation, you must show that it follows the two rules mentioned above. This can be done by using specific examples and showing that the transformation preserves vector addition and scalar multiplication for those examples. Additionally, you can use mathematical proofs to show that the rules hold for all possible inputs.

## 4. What is the difference between a linear transformation and a non-linear transformation?

A linear transformation follows the rules of vector addition and scalar multiplication, while a non-linear transformation does not. This means that a non-linear transformation may not preserve the structure of the original vector space, and the transformed output may not be a vector.

## 5. How are linear transformations used in real life?

Linear transformations are used in many fields, including physics, engineering, and computer graphics. They can be used to model and solve real-world problems, such as predicting the trajectory of a projectile or analyzing the flow of electricity in a circuit. In computer graphics, linear transformations are used to manipulate and transform images and objects in 3D space.

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