How is the following operator linear?

[tamtam402]

How is the following operator linear??!

Homework Statement

Is the following a linear vector function?

F(r) = r - ix

Homework Equations

A function is linear if:

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

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?

 

[Villyer]

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

 

[tamtam402]

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

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).

 

[micromass]

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

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)??

 

[tamtam402]

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)??

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.

 

[tamtam402]

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

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

 

[Mark44]

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

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

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).

 

[algebrat]

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?

 

[vela]

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.

 

[Ray Vickson]

Homework Statement

Is the following a linear vector function?

F(r) = r - ix

Homework Equations

A function is linear if:

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

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?

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

 

[micromass]

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??

 

[Muphrid]

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.