A Linear Algebra Perspective on High School Chemistry

Recently, I was on a ski trip and the topic of chemistry came up, and I realized that high school stoichiometry seemed like a algorithm I should recognize.

In stoichiometry, you're given an equation like this: $$x_1C_6H_{12}O_6 + x_2O_2 = x_3H_2O + x_4CO_2.$$We're trying to determine what the coefficients need to be for the equation to be balanced. From what I remember from high school, you just think really hard until you figure out what they need to be, and that's how you determine the coefficients.

Taking a more systematic approach, we can move everything over to the other side, giving$$x_1C_6H_{12}O_6 + x_2O_2 - x_3H_2O - x_4CO_2 = 0$$

We can consider the balance for each element as an equation, and our coefficients as unknowns$$ \begin{align*} C & &6x_1 + 0x_2 - 0x_3 - 1x_4 = 0\\ H & &12x_1 + 0x_2 - 2x_3 - 0x_4 = 0\\ O & &6x_1 + 2x_2 - 1x_3 - 2x_4 = 0\\ \end{align*} $$

Now it's just a problem of Gaussian elimination. This is a homogeneous linear system:

$$ \begin{bmatrix} 6 & 0 & 0 & -1\\ 12 & 0 & -2 & 0\\ 6 & 2 & -1 & -2\\ \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ x_3\\ x_4\\ \end{bmatrix} = \begin{bmatrix} 0\\0\\0\\ \end{bmatrix}. $$This is so exciting, because now you can use your favorite linear solver on this equation to determine the balance for the original chemical reaction.