Trivial Solution

We like this explanation: “a trivial solution is one that is considered to be very simple and poses very little (or no) academic value.

3a – 6b = 0

You might stare at that and then say a=2, b=1 is one possible solution. You had to put some thought into it.

Another student says, “oh, just put in a zero for the variable in every term, that trick makes any summation sum to zero (and it does). That tricky answer is called a trivial solution.

Consider the summation below where $v_i$ is a vector from a Vector Space and $x_i$ is a scalar from a Field.

$\displaystyle \sum_{i=1}^n x_i v_i = 0$

We avoid doing work (possibly a lot of work) if we settle for the trivial solution: the trivial solution is setting every $x_i[/latex} to zero. Our reason for trying to find a solution to the summation that isn't trivial might be a goal of determining if one vector in the summation equals another vector multiplied by a constant--if yes, then we can set the coefficients for those two vectors in some way where both are nonzero--and reach the goal of the summation equally zero, and if we can do this, it says something important about the set of vectors. Appendix A We haven't found an author to agree with this (yet), but it appears that the word "trivial" can carry with it the notion of "misleading". It seems often the one example that is called "trivial" is the one example that doesn't follow an otherwise universal trend. <ul class="wp-block-list"><li>Adding a number changes a variable except for the trivial case of adding zero.</li><li>Multiplying by a number changes a vaiable except for the trivial case of multiplying by one.</li></ul> A trivial solution might hide something that mathematicians consider to be important. For example, given the differential equation [latex] \dfrac {d} {dx} y = y$

studious work takes us to the solution of $y = e^x$ , and this is valuable for work in several areas of science. The same cannot be said for the trivial solution of $y = 0$ .

Appendix B

The description "a solution which is not interesting" is subjective. Who decides what is "not interesting?" Simplest answer--the teacher who is going to grade your test. We urge you to scrutinize which solutions the teacher calls trivial, and make use of that on the test.

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