how could you prove or disprove that a function is a linear functional?

Question

e.g. that A(L)=x+y is a linear functional defined on the field of complex numbers (C^3) where L=(x) and L is an element of C^3
(y)
(z)

L=(x,y,z) sorry!! can't do matrices with this!

Answer 1

A linear functional is simply a map from the vector space (here C^3) to the underlying field (here C) which is linear; that is, for any u, v in C^3 and and k in C, you must show that
A(u+v) = A(u) + A(v)
and
A(ku) = kA(u).

In his case, let u = (x, y, z) and v = (r, s, t).
Then A(u+v) = A((x+r, y+s, t+z)) (addition in C^3)
= (x+r) + (y+s) (definition of A)
= (x+y) + (r+s) (field properties of C)
= A(u) + A(v) (definition of A)

and A(ku) = A(kx, ky, kz) (scalar multiplication in C^3)
= kx + ky (definition of A)
= k(x + y) (distributive law for C)
= kA(u) (definition of A).

Hence A is a linear functional.

To disprove something is a linear functional, you need to find an example of a pair of vectors u and v such that A(u+v) does not equal A(u) + A(v), or a vector u and a constant k such that A(ku) does not equal kA(u). This usually isn't hard.

Answer 2

Scarlet is on the right track. You can also check (to disprove) if A(0) is not equal to 0 because A(0)=0 for any linear functional. NOTE: there is a difference between a linear function (other posters) and linear functional.

Answer 3

I remember from Pre-Calculus that a set of values such as (3,5,9,11,) and another set of values ( 33,39,71,93) can be evaluted using the the fromula for slope. Take the first number from each set and treat it like an ordered pair. Take the second set and treat it like another ordered pair, and plug them into the formula for slope Y2 - Y1 / X2 -X1 . then repeat the process for Y3 - Y2 / X3 - X2 and repeat for Y4 - Y3 / X4 - X3. if all the slopes are equal you have a linear function, otherwise it's Non-linear

Answer 4

any function has form Ax + By + C = 0 is a linear function

Answer 5

Put in a series of values for your variables and plot the outcomes on a graph. The result will either be a line or not.

Answer 6

When graphed or charted, it will produce a straight line if it is linear