What makes something linear

A non-linear equation is such which does not form a straight line. It looks like a curve in a graph and has a variable slope value.

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Here, x is a variable, and a and b are constants. Linear Equation Examples.

So the linear equation can be thought of very much in terms of a line. The slope and one point on the line is all that is needed to write the equation of a line. There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form. Standard form is a way of writing down very large or very small numbers easily. Therefore we can conclude that the problem has no solution.

If the graphs of the equations do not intersect for example, if they are parallel , then there are no solutions that are true for both equations. If the graphs of the equations are the same, then there are an infinite number of solutions that are true for both equations. An infinite solution has both sides equal. A system of linear equations can have no solution, a unique solution or infinitely many solutions. A system has no solution if the equations are inconsistent, they are contradictory.

An equation can have infinitely many solutions when it should satisfy some conditions. The system of an equation has infinitely many solutions when the lines are coincident, and they have the same y-intercept. If the two lines have the same y-intercept and the slope, they are actually in the same exact line. Look at option C. Is it intended to be a distractor? Will a student who chooses it be marked wrong? The moral of this story is, I suppose, that it is easier to tell when a function is not linear than to tell when it is linear.

Testing for non-linearity involves just picking a few points on the graph; testing for linearity involves picking every possible pair of points on the graph and verifying that the slope between them is always the same. When talking about a "bounded linear functional" or, equivalently, a continuous linear functional , I always think of lines pinned at zero with varying slope.

If you have an unbounded linear functional, things get weird. These guys do not have a nice representation theorem like Hahn-Banach. I always try to remember two things about this class of operator: 1 they can only appear in infinite-dimensoinal spaces. A linear code or algorithm is a rather broad notion. It could be confusing, since a "linear algorithm" might do a task that is nonlinear or hard to even formulate in a vector space. This is in contrast with algorithms which may require a different amount of time to complete, e.

If I'm writing a code, and visually I can see nested for-loops, then most of the time the code is not linear. If I hear someone say "We have a linear algorithm to compute X" then I'm usually impressed, since it means they found a pretty efficient way to complete the task X.

Sometimes an object called linear if there is no loop or circle. This is something like this 2 phase logic: if it is not p then it is q. A function called linear if it preserve linearity of a linear object. It seems plausible for me to assume that you are refering to such functions that are linear "predictors". Well, to answer your question on the things that follow the word linear, I think it just means operating in one dimension, relating to lines or just that the specific thing is "simple".

Or, a linear system is something that doesn't just exist in math, but in real visual space. And something that you can physically interact with. If I were asked for an elevator pitch about the concept of "linear" I would start with the better known concept of proportionality. You can fill out that idea with different kinds of mathematical examples, depending on the level of the elevator mates you are enlightening. That covers the examples of linear differential equations when you rewrite them as differential operators on a vector space of functions.

In fact, vector spaces were invented to provide the abstract framework for "linearity", which is captured by the ability to add vectors and multiply them by scalars. So when I see a reference to a "linear something" I am pretty sure there's a vector space lurking.

Linear codes are not examples of proportionality which deals with the scalar multiplication but are examples where the ability to add vectors comes into play: code words can be thought of as vectors, and the vector sum of codewords is again a codeword. It's often useful when thinking about a new concept to consider things that aren't it. So let me draw-- I'll do a rough graph here. So let me make that my vertical axis, my y-axis.

And we go all the way up to So I'll just do 10, 20, Actually, I can it do a little bit more granularly than that. I could do 5, 10, 15, 20, 25, 30, and then And then our values go 1 through 5.

I'll do it on this axis right here. They're not obviously the exact same scale, so I'll do 1, 2, 3, 4, and 5. So let's plot these points. So the first point is 1, 11, when x is 1, y is This is our x-axis.

When x is 1, y is 11, that's right about there. When x is 2, y is 14, that's right about there.