Why is a one-variable linear equation called linear?

[gullpacha]

TL;DR Summary

why one variable linear equation called linear?

i get preparation for my university entering exam and i studying linear equation and they define it linear equation are those which graph line now my question is that why one variable linear equation called linear for example y=3 that actually give us just line why we draw line for this equations?
2- is 10/x=2 is linear equation and is square root of (y-9)=10 is linear if no why these equation are not linear?

 

[fresh_42]

This is partly a bit of abuse of notation. The functions $x\longmapsto c\cdot x$ are linear. That we call $x\longmapsto c\cdot x+b$ linear, too, is the abuse. The latter is affine linear because linear functions map $0$ to $0$ and if $b\neq 0$ then it's graph is still a straight line (hence 'linear'), but the shift $0\longmapsto b$ makes it mathematically affine linear.

Later on in differential geometry this differences vanishes further: If we say a tangent $t$ is linear, then it means: $t=p+ \mathbb{R}^1$, which is strictly spoken affine linear due to $p$. However, if we only consider this tangent space, then we identify $p$ with the origin of $\mathbb{R}^1$ which thus makes $t$ linear. And if we consider the derivative as linear function, then we mean $x\longmapsto \left. \dfrac{df}{dx}\right|_{p}\, \cdot \, x$.$^*)$

$^*)$ This is only true in dimension one where we have only one direction in which we differentiate. If there are more directions possible, then the tangent as linear function is best seen in the Weierstraß notation:
$$\mathbf{f(p+v)=f(p)+J(v)+r(v)}$$
where the linear function is the Jacobi matrix $J$ and $v$ the direction of differentiation of $f$, and $r$ the error we made by approximation of $f(p+v)$ by $f(p)+J(v).$

 

[mfb]

@fresh_42: That's far above [B] level.

Summary:: why one variable linear equation called linear?

i get preparation for my university entering exam and i studying linear equation and they define it linear equation are those which graph line now my question is that why one variable linear equation called linear for example y=3 that actually give us just line why we draw line for this equations?

Everything that can be written as y=ax+b is considered linear here. More strictly, it's y depending on x in a linear way. y=3 can be written in that way: y=0*x+3. This assumes that y is something that could (in general) depend on x.

is 10/x=2 is linear equation

You could see it as "x depends on y (or any other variable) in a linear way, but that would be a stretch. Usually x is used for independent variables. An equation that gives you x would be outside the linear/nonlinear classification.

and is square root of (y-9)=10 is linear

In the real numbers this is equivalent to y=109. After a transformation you have the same situation as in the first case.

Ultimately all that doesn't really matter. It's just a name. If the name could be ambiguous then be more explicit what you mean. As simple as that.

 

[Tom.G]

One definition of the word LINEAR is 'something that is straight.'
Following that, an equation that plots as a straight line is called 'Linear.'

Summary:: why one variable linear equation called linear?

is 10/x=2 is linear equation and is square root of (y-9)=10 is linear if no why these equation are not linear?

Neither of those equations plot as a straight line, so no, they are not linear.

The only common operations I can think of to generate a straight line are Addition, Subtraction, and Multiplication of two variables.

Hope this helps!

Cheers,
Tom

p.s. there may be more linear operations in higher math, if so they do not belong in a 'B' thread.

 

[Office_Shredder]

Tom, I think both those equations do draw straight lines (one vertical, one horizontal).

 

[phinds]

Tom, I think both those equations do draw straight lines (one vertical, one horizontal).

Except that I would leave out the "I think that".

 

[symbolipoint]

A simple-minded way to think is an equation with any and all variables to power of 1, no ratio of any variables used, is linear. @mfb seems to give the right idea.

 

[DaveE]

In my experience (engineering), there are two different definitions of "linear" functions in common use.
The first, which most learn in their first algebra class, is an equation that describes points that lie on a line.
The second, more common in systems analysis is that a function f is linear iff f(a+b) = f(a) + f(b).
Of course, they aren't the same, as the OP's example shows.

 

[etotheipi]

The second, more common in systems analysis is that a function f is linear iff f(a+b) = f(a) + f(b).

It must also preserve the scalar multiplication, $f(kx) = kf(x)$

 

[weirdoguy]

It must also preserve the scalar multiplication

Yes, but that is called homogeneity of $f$: https://en.wikipedia.org/wiki/Homogeneous_function

 

[member 587159]

Yes, but that is called homogeneity of $f$: https://en.wikipedia.org/wiki/Homogeneous_function

@etotheipi is not wrong. Linearity refers (in the context of abstract algebra) to both the preservation of the scalar multiplication and the addition. A linear map is a map that is both homogeneous and additive.

 

[weirdoguy]

@etotheipi is not wrong.

I know, that's why I started my sentence with "yes" But overall it was ambiguous, I'm sorry.

 

[DaveE]

It must also preserve the scalar multiplication, $f(kx) = kf(x)$

But it has to if it satisfies superposition. f(a+a+a+...) = f(a) + f(a) + f(a) + ...

edit: OK now I'm not so sure. Why does Wikipedia ("linear function") require both? Is there a function that has superposition but isn't homogeneous?

 

[Office_Shredder]

I think you can construct strange looking everywhere-discontinuous functions that aren't linear but are additive.

For example, imagine R as an infinite dimensional vector space over Q. Pick a basis (I think you can pick a basis for this vector space) Pick a Q-linear function that maps one basis element to zero, and acts as the identity function on the rest of the basis vectors. This function is additive over R, but not linear on R when viewing it as a one dimensional vector space over R.

 

[mfb]

While that should work you can construct a much simpler example in the complex numbers. f(z)=Re(z)+Im(z) satisfies f(a+b)=f(a)+f(b), but $i f(1) \neq f(1i)$

 

[Office_Shredder]

That is a much nicer example.

 

[Tom.G]

Summary:: why one variable linear equation called linear?

10/x=2... square root of (y-9)=10

Tom, I think both those equations do draw straight lines (one vertical, one horizontal).

Interesting! Can you specify 3 values of the variables in each of those that, as written, would plot as a straight line in rectilinear 2-space?

I see the first one plotting as a point. The second as a point, or possibly as two points depending on whether the duality of SQR in considered.

Cheers,
Tom

 

[Office_Shredder]

Tom, 10/x=2 is equivalent to x=5. So some valid points would be (5,0), (5,2), (5,8) etc.

sqrt(y-9)=10 is the same as y-9=100, y=109. So some points are (0,109), (-2,109), (12,109) etc.

 

[Tom.G]

I see, we are using different interpretations of the question.

I was considering each of the variables as a discrete number (on the number line), whereas you were considering the variables as the universe of lines that pass thru a point on the plane.

Equivalently, I used a literal interpretation, as the OP presented the question. As opposed to the approach which first solved for the variable then included the universe of solutions.

Fair enough! And thanks.

Cheers,
Tom