Linear transformation proof?

1. Mar 14, 2013

dylanhouse

1. The problem statement, all variables and given/known data

See attached image below.

2. Relevant equations

3. 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.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Attached Files:

• Linear A. Question.jpg
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2. Mar 14, 2013

Staff: Mentor

It's not very clear from the description but the transformation could also be "desribed" this way: f : R → M2x2,
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).

3. Mar 14, 2013

LCKurtz

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

4. Mar 14, 2013

LCKurtz

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

5. Mar 14, 2013

Fredrik

Staff Emeritus
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.

6. Mar 14, 2013

dylanhouse

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

7. Mar 14, 2013

Staff: Mentor

What I wrote is how I interpreted the OP's attachment.
Yes. It was unclear to me, as well.

8. Mar 14, 2013

dylanhouse

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

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9. Mar 14, 2013

Fredrik

Staff Emeritus
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).

10. Mar 14, 2013

Fredrik

Staff Emeritus
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.