Do linear theories have infinitely many solutions?

Question

It is said that, in a linear theory, you can add any number of solutions to still find a solution. So, say I initially found three solutions to a linear theory, and I call them a1, a2, and a3. Now I can add a1 and a2 and call the sum a4 (a1+a2=a4), so a4 also becomes a solution. Then I can add a4 to a1 so a4+a1=a5 becomes a solution. This way, it becomes like a chain reaction and can reach upto infinity. Does it mean all linear theories have infinite number of solutions?
Examples of linear theories can be Maxwell’s theory of electromagnetism, Quantum Mechanics, etc.

mmesser314 · Answer

Solutions of a linear theory are a vector space. Vectors are things that can be added together and multiplied by numbers.
If $a_1$ is a solution, so is $2a_1$, $3a_1$, etc. If $a_1$ and $a_2$ are solutions, so are $Aa_1 + Ba_2$ for any numbers $A$ and $B$.
There are always an infinite number of vectors in a vector space. (OK, except for the trivial $0D$ vector space.)
But if you have two different solutions that are not multiples of each other, it does not generate a chain reaction of solutions. It generates a $2D$ vector space of solutions.

Do linear theories have infinitely many solutions?

One Answer

Add your own answers!

Ask a Question