• fredrogers3
In summary, the question is about deciding if given transformation functions are linear or not. The first three transformations are not linear because they do not satisfy the constraint A(cx+dy) =c(Ax)+d(Ay). Only the last transformation is linear with the associated matrix of transformation [1 -2]. For the first transformation, A is defined as Ax = x1 + x2 + 3, and for the other two transformations A is not defined as it is impossible to write an associated matrix of transformation.
fredrogers3

## Homework Statement

Hello everyone,

I have a quick question about linear transformations. In my class, we were given transformation functions and asked to decide if they are linear:

The transformation defined by: T(X)= X1+X2+3
The transformation defined by: T(X)=X1+X2+(X1*X2)
The transformation defined by: T(X)= 2X1*X2
The transformation defined by: T(X)= X1-2X2

For those that are not linear transformations, we were asked to state why they were not a linear transformation.

See Below

## The Attempt at a Solution

I correctly figured out that the only linear transformation on the list was the last transformation. This has associated matrix of transformation [1 -2]. My rationale for not choosing the other equations was the fact that it is impossible to write an associated matrix of transformation for the T(X). This was done by simple inspection. However, I was told this is not an acceptable justification for why those are not linear. I thought that all linear transformations must have an associated matrix.

Further, my book says that all transformations that satisfy A(cx+dy) =c(Ax)+d(Ay) are linear. However, it doesn't seem that the first 3 equations satisfy this constraint.
Any thoughts?

Thanks

fredrogers3 said:

## Homework Statement

Hello everyone,

I have a quick question about linear transformations. In my class, we were given transformation functions and asked to decide if they are linear:

The transformation defined by: T(X)= X1+X2+3
The transformation defined by: T(X)=X1+X2+(X1*X2)
The transformation defined by: T(X)= 2X1*X2
The transformation defined by: T(X)= X1-2X2
I presume you mean that X= (X1, X2, X3). You should say that.

For those that are not linear transformations, we were asked to state why they were not a linear transformation.

See Below

## The Attempt at a Solution

I correctly figured out that the only linear transformation on the list was the last transformation. This has associated matrix of transformation [1 -2]. My rationale for not choosing the other equations was the fact that it is impossible to write an associated matrix of transformation for the T(X). This was done by simple inspection. However, I was told this is not an acceptable justification for why those are not linear. I thought that all linear transformations must have an associated matrix.

Further, my book says that all transformations that satisfy A(cx+dy) =c(Ax)+d(Ay) are linear. However, it doesn't seem that the first 3 equations satisfy this constraint.
Any thoughts?

Thanks
Yes, A is a linear transformation if and only if A(cx+ dy)= cA(x)+ bA(y) for a and d numbers and x and y vectors.

And, yes, only the last is a linear transformation. But can you say exactly why that s true?

In the first, A is defined by Ax= A(x1, 2)= x1[//sub]+ x2+ 3.

So A(ax+ by)= A(ax1+ by1, ax2+ by2)= ax1+ by1+ ax2+ by2+ 3.

While aA(x)= a(x1+ x2) while bA(y)= b(y1+ y2

Thank you for the help! Just to clarify, would the correct matrix of transformation for the 4th equation be [1 -2]?

## 1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the underlying structure of the space. In simpler terms, it is a transformation that can be represented by a matrix and follows certain rules, such as preserving the lines and origin of the original space.

## 2. How do you determine if a transformation is linear?

To determine if a transformation is linear, you can check if it follows two main rules: preserving addition and scalar multiplication. This means that for any two vectors u and v and scalar c, the transformation of u + v should equal the transformation of u plus the transformation of v, and the transformation of cu should equal c times the transformation of u.

## 3. What are some examples of linear transformations?

Some common examples of linear transformations include rotations, reflections, and scaling. Other examples include projection, shearing, and translation. In general, any transformation that can be represented by a matrix and follows the rules of linearity is considered a linear transformation.

## 4. How do you perform a linear transformation on a vector?

To perform a linear transformation on a vector, you can simply multiply the vector by the transformation matrix. The resulting vector will be the transformed version of the original vector. It is important to note that the dimensions of the vector and the transformation matrix must match in order for the multiplication to be possible.

## 5. What is the importance of linear transformations in science?

Linear transformations are important in science because they provide a way to model and understand real-world phenomena. Many physical systems and processes can be described and analyzed using linear transformations, making them a powerful tool in fields such as physics, engineering, and computer science.

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