How is the following operator linear?

You can verify that it satisfies the linear operator properties you gave earlier.In summary, the conversation discusses whether the given operator, F(r) = r - ix, is a linear vector function. The definition of a linear function is provided, and it is determined that F(r) is not linear in the traditional sense. However, it is pointed out that F(x,y,z) = (0,y,z) can be seen as a linear operator. A revised expression for the given operator is also provided, which satisfies the linear operator properties.
  • #1
tamtam402
201
0
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?
 
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  • #2


Well you would agree that f(x) = mx + b is linear, right?
But f(x1 + x2) != f(x1) + f(x2)
 
  • #3


Villyer said:
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).
 
  • #4


Villyer said:
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 [itex]\mathbf{r}[/itex]? What is x?? What is [itex]\mathbf{i}[/itex] (I suspect it is just the first basis vector)?

Could it be that [itex]\mathbf{r}=(x,y,z)[/itex]??
 
  • #5


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

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

Could it be that [itex]\mathbf{r}=(x,y,z)[/itex]??

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


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

f(r1) +f(r2) = 2jy + 2kz
 
  • #7


tamtam402 said:
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).
 
  • #8


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


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


tamtam402 said:

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
 
  • #11


You may want to write your F as

[tex]F(x,y,z)=(0,y,z)[/tex]

Do you agree that this is your F?? Do you see that this is linear??
 
  • #12


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

[itex]\underline F(a) = a - (a \cdot i) i[/itex]

which clearly is linear.
 

1. What is the definition of a linear operator?

A linear operator is a mathematical function that maps one vector space to another while preserving the properties of addition and scalar multiplication. In other words, if f is a linear operator, then for any two vectors x and y and scalar c, f(x + y) = f(x) + f(y) and f(cx) = cf(x).

2. How can I determine if an operator is linear?

To determine if an operator is linear, you can use the definition of a linear operator. If the operator satisfies the properties of preserving addition and scalar multiplication, then it is linear. You can also check if the operator follows the rules of linearity, such as f(x + y) = f(x) + f(y) and f(cx) = cf(x).

3. What are some common examples of linear operators?

Some common examples of linear operators include differentiation and integration operators, matrix multiplication, and the Laplace operator. These operators preserve the properties of addition and scalar multiplication, making them linear.

4. Can an operator be linear in one case but not in another?

Yes, an operator can be linear in one case but not in another. For example, a differentiation operator is linear when applied to polynomials, but it is not linear when applied to trigonometric functions. This is because the derivative of a sum is the sum of the derivatives, but the derivative of a product is not the product of the derivatives.

5. How is the concept of linearity used in real-world applications?

The concept of linearity is used in many real-world applications, such as engineering, physics, and economics. Linear operators are used to model and solve problems involving systems that exhibit linear behavior, such as electrical circuits, chemical reactions, and fluid flow. They are also used in data analysis and machine learning to make predictions and identify patterns in data.

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