# 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.

#### Attached Files:

• ###### Algebra.jpg
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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.