Is a linear equation always a straight line?

Question

I am studying Linear Algebra, and it just blows my mind how so many things in real life can be related to linear equations. I know this might be a stupid question, but is it correct to assume that all linear equations are just straight lines (with or with out slopes?), that none of the equations will ever have exponents, logarithms, etc.??

in this link, it uses an example with a variable being squared. how is that a linear equation?

I understand that you can have planes in 3 Dimensional spaces, but I guess my main question is, will there ever be a curve in a linear equation?

Linear algebra deals with the study of linear systems. A "linear system" is a system of lines. So yes, when applying matrices, a linear system is involved. That being said, there are typically ways to transform non-linear problems to linear problems, so linear algebra can be applied to a variety of problems. — Kaynex, Mar 31 '17 at 19:11
@Kaynex What would be an example of this? Is there a theorem I can read about that transforms curved lines into linear equations that you know of? — CtrlAltDelete, Mar 31 '17 at 19:14
A very common application of this is differential equations. Solving a set of dynamic systems is made easier by using a good transform and linear algebra. Complicated problems in classical mechanics get simplified with the methods taught in linear algebra. — Kaynex, Mar 31 '17 at 19:35
You can use substitution to turn curved systems into linear systems. For example, $$1/x + 2/y = 2$$ $$2/x + 1/y = 3$$ can be solved with $u = 1/x$ and $v = 1/y$ — Kaynex, Mar 31 '17 at 19:39

score 1 · Answer 1 · answered Mar 31 '17 at 19:07

1

If your vector space is $\mathbb{R}^2$, then a linear equation in the variables $x$ and $y$ is effectively a line. On the contrary, when you pass in $\mathbb{R}^3$, an equation of the form $ax+by+cz+d=0$ represents a plane. In order to obtain a straight line, you must consider two linear equations $ax+by+cz+d=0$ and $a'x+b'y+c'z+d'=0$ so that $(a,b,c)$ and $(a',b',c')$ are non proportional.

answered Mar 31 '17 at 19:07

TheWanderer

5,126

So if you are in a 3D space, you can now have curves? – CtrlAltDelete Mar 31 '17 at 19:08
$x^2+y^2=1$ is the equation of a circle in the plane, so you have curves also in 2D space. – TheWanderer Mar 31 '17 at 19:46

score 1 · Answer 2 · answered Mar 31 '17 at 19:10

1

The zero locus of linearly related variables (by $f(x,y)=ax+by$ in the below) in $\mathbb{R}^2$ is a line through the origin. Meaning, the set of all $x$ and $y$ with $$ f(x,y)=ax+by=0 $$

This is a fancy way of saying that the graph of a linear map from $R^1$ to $R^1$ is a line through the origin.

answered Mar 31 '17 at 19:10

operatorerror

29,103

Okay but in any dimension can anything ever be curved? – CtrlAltDelete Mar 31 '17 at 19:11
@CtrlAltDelete not if the map is linear. Is the plane in the above example curved? – operatorerror Mar 31 '17 at 19:11
For example a line has no curvature, a plane has no curved edges, a cube has no curved edges, etc. Can you ever have a circle a sphere? – CtrlAltDelete Mar 31 '17 at 19:12
no.$\quad{}{}{}{}$ – operatorerror Mar 31 '17 at 19:12
what is an example of a non linear map? – CtrlAltDelete Mar 31 '17 at 19:13
@CtrlAltDelete $f(x)=x^2$. But these are things you can check on your own. Take some map and check to see whether $f(ax+by)=af(x)+bf(y)$ – operatorerror Mar 31 '17 at 19:13
Let us continue this discussion in chat. – CtrlAltDelete Mar 31 '17 at 19:15

score 0 · Answer 3 · edited Apr 13 '17 at 12:20

From Wiki:

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

This does not mean that we can't use nonlinear tools and concepts to help study these objects. If that's too many negatives in one sentence, another way to say it is it's perfectly fine to use nonlinear tools to study linear objects.

As for your questions...

but is it correct to assume that all linear equations are just straight lines (with or with out slopes?)

Yes. But once you go higher than 3 dimensions it becomes sort of... difficult to visualize. For example, $2x_1 + 3x_2 - x_3 + 7x_4 = 0$ technically represents a line in 4 dimensions. The concept of 2-dimensional slope is more or less generalized to higher dimensions with the gradient.

that none of the equations will ever have exponents, logarithms, etc.??

No. For example, $\pi^5 x + (\ln 2) y = 0$ is a linear equation containing exponents and logs. But $\pi x^5 + 2\ln y = 0$ is not a linear equation, which may be what you were getting at.

in this link, it uses an example with a variable being squared. how is that a linear equation?

I'm assuming you're talking about $y = \alpha x^2 + \beta x + \gamma$ in this answer to the question you linked to. That's not a linear equation. The answer author even said it's a second-order polynomial function of $x$. This is an example of what I originally mentioned, i.e., using nonlinear tools to study linear objects. In this case the linear object being studied is what the answer author goes on to talk about, which is a system of 100 equations in 3 unknowns. The unknowns are the constants $\alpha, \beta, \gamma$. And yes, that system does include $x_i^2$ in the matrix $X$, but the equations in the system are still linear because the variables in the system are $\alpha, \beta, \gamma$, not $x$. It's the coefficients in the system that have the exponents, which is fine.

I understand that you can have planes in 3 Dimensional spaces, but I guess my main question is, will there ever be a curve in a linear equation?

No, not in Euclidean geometry. (I think lines are technically curved in some types of non-Euclidean geometry but that may be getting too off topic.)

Is a linear equation always a straight line?

3 Answers3