Proving Linearity of a Transformation: Where to Start?

[dylanhouse]

Homework Statement

See attached image below.

Homework Equations

The Attempt at a Solution

I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

 

[Mark44]

Homework Statement

See attached image below.

Homework Equations

The Attempt at a Solution

I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

It's not very clear from the description but the transformation could also be "desribed" this way: f : R → M_2x2,
where
$$ f(c) = \begin{bmatrix} 1 & c \\ 0 & 1\end{bmatrix}$$

Start by calculating f(x), f(y), and f(x + y) and seeing if f(x) + f(y) = f(x + y). Then check that f(tx) = t*f(x).

 

[LCKurtz]

Homework Statement

I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

Homework Statement

Do you understand what the domain of the transformation is and its "formula" as it acts on the domain?

 

[LCKurtz]

It's not very clear from the description but the transformation could also be "desribed" this way: f : R → M_2x2,
where
$$ f(c) = \begin{bmatrix} 1 & c \\ 0 & 1\end{bmatrix}$$

How do you know it isn't a transformation from $R^2$ to $R^2$?

 

[Fredrik]

Homework Statement

See attached image below.

Homework Equations

The Attempt at a Solution

I know for it to be a linear transformation it must be that: f(x)+f(y)=f(x+y) and f(tx)=tf(x) where t is a scalar. I'm not sure where to start with this proof.

You didn't actually post the definition of $f_A$. The image says that $f_A$ is "described" by that matrix, but what does that mean?

It's not very clear from the description but the transformation could also be "desribed" this way: f : R → M_2x2,
where
$$ f(c) = \begin{bmatrix} 1 & c \\ 0 & 1\end{bmatrix}$$

Start by calculating f(x), f(y), and f(x + y) and seeing if f(x) + f(y) = f(x + y). Then check that f(tx) = t*f(x).

I don't think this is right. The problem said we should check that $f_A$ is linear for all c. So, there should be one function for each value of c. The function you called f is just one function. It's more likely that we should check that what you called f(c) is linear, i.e. that for all real numbers c, we have f(c)(ax+by)=af(c)x+bf(c)y for all vectors x,y and all real numbers a,b.

The OP should explain how $f_A$ is defined.

 

[dylanhouse]

I'm not sure. We only did a little on matrix transformations. Would f(x) be that matrix with x instead of c?

 

[Mark44]

How do you know it isn't a transformation from $R^2$ to $R^2$?

What I wrote is how I interpreted the OP's attachment.

You didn't actually post the definition of $f_A$. The image says that $f_A$ is "described" by that matrix, but what does that mean?



I don't think this is right. The problem said we should check that $f_A$ is linear for all c. So, there should be one function for each value of c. The function you called f is just one function. It's more likely that we should check that what you called f(c) is linear, i.e. that for all real numbers c, we have f(c)(ax+by)=af(c)x+bf(c)y for all vectors x,y and all real numbers a,b.

The OP should explain how $f_A$ is defined.

Yes. It was unclear to me, as well.

 

[dylanhouse]

This is the original question, as stated on the handout.

 

[Fredrik]

OK, but the notation $f_A$ where A is a matrix, or the concept of a function being "described" by a matrix, should be explained somewhere in the book.

I would guess that you're supposed to show that regardless of the value of c, the function $f_A:\mathbb R^2\to\mathbb R^2$ defined by $f_A(x)=Ax$ for all $x\in\mathbb R^2$, is linear. (It is however a little bit weird to ask for this, since these functions are linear for all 2×2 matrices A, not just the ones mentioned in the problem. So you should still try to find a definition in your book or in some other handout).

 

[Fredrik]

The statement of problem 8 (which you posted here) strongly suggests that $f_A$ is to be interpreted the way I did in my previous post.