# Homework Help: How is the following operator linear?

1. Jun 9, 2012

### tamtam402

How is the following operator linear??!

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

Is the following a linear vector function?

F(r) = r - ix

2. Relevant equations

A function is linear if:

F(r1 + r2) = F(r1) + F(r2) AND F(ar) = aF(r)

3. The attempt at a solution

F(r1 + r2) = (r1 + r2) - ix

F(r1) + F(r2) = (r1 - ix) + (r2 - ix) = (r1 + r2) - 2ix

According to me, F(r1 + r2) ≠ F(r1) + F(r2), but Mary L Boas says otherwise. What gives?

Last edited: Jun 9, 2012
2. Jun 9, 2012

### Villyer

Re: How is the following operator linear??!

Well you would agree that f(x) = mx + b is linear, right?
But f(x1 + x2) != f(x1) + f(x2)

3. Jun 9, 2012

### tamtam402

Re: How is the following operator linear??!

I'm not sure I understand your point? Yes, f(x) = mx + b is the function of a line, but it is NOT a linear function, since f(x1 + x2) ≠ f(x1) + f(x2). I'm not sure where you're getting at.

According to the Mary L Boas book, the operator I posted is supposed to be a linear operator, which means f(r1 + r2) = f(r1) + f(r2) and f(kr) = kf(r). My results show otherwise, since I find f(r1 + r2) != f(r1) + f(r2).

4. Jun 9, 2012

### micromass

Re: How is the following operator linear??!

f(x)=mx+b is not linear.

OP: could you quote the entire problem. What is $\mathbf{r}$? What is x?? What is $\mathbf{i}$ (I suspect it is just the first basis vector)?

Could it be that $\mathbf{r}=(x,y,z)$??

5. Jun 9, 2012

### tamtam402

Re: How is the following operator linear??!

i is the first basis vector, and r has been defined earlier in the book as (x,y,z), or ix + jy + kz.

I must be missing something, because it seems (according to your post) that r being (x,y,z) would change my solution, but I don't see how.

6. Jun 9, 2012

### tamtam402

Re: How is the following operator linear??!

I still find f(r1+r2) = ix + 2jy + 2kz

f(r1) +f(r2) = 2jy + 2kz

7. Jun 9, 2012

### Staff: Mentor

Re: How is the following operator linear??!

You aren't distinguishing between the components of r1 and r2.

Let r1 = (x1, y1, z1), and r2 = (x2, y2, z2).

Now calculate f(r1 + r2) and compare that to f(r1) + f(r2).

8. Jun 9, 2012

### algebrat

Re: How is the following operator linear??!

Is this just a question about the multiple definitions of linear?

y=Ax+b is linear in one sense, and not linear in another sense, right?

9. Jun 9, 2012

### vela

Staff Emeritus
Re: How is the following operator linear??!

Right. y=mx+b is the equation of a line in the xy-plane, so it's often called linear in that sense. It is not, however, linear in the linear algebra sense, which is what the OP's question is about.

10. Jun 9, 2012

### Ray Vickson

Re: How is the following operator linear??!

What gives is that your x needs to be tied to your r, so if you have r1 and r2 you need x1 and x2.

RGV

11. Jun 9, 2012

### micromass

Re: How is the following operator linear??!

You may want to write your F as

$$F(x,y,z)=(0,y,z)$$

Do you agree that this is your F?? Do you see that this is linear??

12. Jun 9, 2012

### Muphrid

Re: How is the following operator linear??!

To make it clear, I think the proper expression of your linear operator is this:

$\underline F(a) = a - (a \cdot i) i$

which clearly is linear.