how do you prove that something is linear?

Question

I'm taking linear algebra...i need to know how to prove if a function is linear. How do i go about it?

Answer 1

You need to show two conditions:

If f: V --> W is a map between vector spaces V and W over the real numbers R (over any field F), then f is linear if and only if

1) f(v_1 + v_2) = f(v_1) + f(v_2)

2) f(av) = af(v)

for all v, v_1, v_2 ∈ V and all a ∈ R.

Here are two examples.

1) Let V = W = R^2 and let f : V --> W, (x,y) |--> (x + 3y, -y).
We choose two arbitrary vectors v_1 and v_2 of V. In coordinate form, we write v_1 = (x_1, y_1) and v_2 = (x_2, y_2). Then:

f(v_1 + v_2) = f((x_1,y_1) + (x_2,y_2)) = f((x_1 + x_2, y_1 + y_2)) = (x_1 + x_2 + 3(y_1 + y_2), -(y_1 + y_2)) = (x_1 + x_2 + 3y_1 + 3y_2, -y_1 - y_2).

On the other hand, f(v_1) + f(v_2) = f((x_1,y_1)) + f((x_2,y_2)) = (x_1 + 3y_1, -y_1) + (x_2 + 3y_2, -y_2) = (x_1 + x_2 + 3y_1 + 3y_2, -y_1 - y_2).

Now let v = (x,y) be another vector in V and let a be a real number. Then

f(av) = f(a(x,y)) = f((ax, ay)) = (ax + 3ay, -ay)

and

af(v) = af((x,y)) = a(x + 3y, -y) = (a(x + 3y), a(-y)) = (ax + 3ay, -ay).

Therefore, f is linear.

2) Let V = W = R^2, f : V --> W, (x,y) |--> (x^2, y).

To show that a function is not linear, we only need to find an example of where one of the above two conditions break down. Take v = (1,0), a = 2. Then

f(2(1,0)) = f((2,0)) = (4,0)

but

2*f((1,0)) = 2*(1,0) = (2,0).

Therefore, f is not linear.

Answer 2

For a function to be linear, for example if y is a function of x or y=f(x), y and x both must have a power of 1.

For example:

y = 4x+35 (i.e. y^1 = 4x^1 + 35)

5y-2.5x = 6.6x+23.1y+4

So y or x must be raised to the power of 1 meaning they must not be squared or raised to any other power than 1.

So if we plot x against y on a graph we will see a line if x or/and y are for example squared or cubed then the function would be curved.

Note that the square root is a power; y = sqrt(x) (i.e y = x^(1/2)) is also a curve function.

Of course if you want to prove to yourself that it's a linear function (so its slope must be constant since every line has a constant slope or inclination angle) you can get the first derivative dy/dx and if it turned out to be a constant number and didn't include x then this function is linear.

For example:

y = 4x+35

then,
dy/dx = 4 which means that the tangent of the angle between the line and the x axis is 4.

So the angle would be arctan(4)

If the slope is positive the line is inclined to the right if it's negative then the line is inclined to the left (assuming that the positive direction of the x-axis is to the right).

Answer 3

If the slope of a line is constant, the line is linear.

best way to check this is to make a table and divide your output by the input (y/x). If you keep getting the same answer, the line is linear (a straight line).

Answer 4

There are many ways. One way is to prove that the gradient is constant. Another is to prove that the given function f(x) differs from a straight line by a constant amount for all x.

Answer 5

The formula will have a straight line equation (y = mx + c).
Any value taken for y must equal mx + c, and any x value put into the equation, multiplied by m and c added must equal y.

Answer 6

If you graph it you should come up with a line as a result

Answer 7

make sure there are no polynomials in your equation

Answer 8

en.wikipedia.org/wiki/Linear_equation

Please click on the above URL for further information.