Question about Linear Transformations

[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.

Homework Equations

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

 

[HallsofIvy]

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.

Homework Equations

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)= x<sub>1[//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</sub>

 

[fredrogers3]

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