Vector Equations

A vector equation is simply a system of equations read column by column instead of row by row: it asks whether you can reach a target vector by taking some amount of each of a few other vectors and adding them up. Think of directions like "walk 3 blocks east, then 2 blocks north"; the question is how many blocks of each. That's good for any mixing problem, like how many servings of two foods give a target amount of protein and carbs. The equation $x_1\vec a_1 + x_2\vec a_2 + \cdots + x_n\vec a_n = \vec b$ looks heavy, but it has only three kinds of pieces: $\vec a_1, \dots, \vec a_n$ (the directions you're allowed to move in, which are the columns of your system), $x_1, \dots, x_n$ (how far to go in each direction, the weights you're solving for) and $\vec b$ (where you want to end up). It's the same system as before, just seen as a picture; bear with it.

From systems to vectors. Take a look at the system in the worked example below: $x_1 + 3x_2 = 7$ and $2x_1 - x_2 = 4$. Stack it up by columns and it says

Nothing new happened: the top entries give the first equation and the bottom entries give the second. But now there's a picture. You're combining two arrows to land on a third.

Linear combinations. $3\vec u - 2\vec v$ means: walk 3 copies of $\vec u$, then 2 copies of $\vec v$ backward. Any sum of scaled vectors like that is called a linear combination.

Span is everywhere such walks can reach. One nonzero vector spans a line through the origin, since all you can do is go forward or back along it. Two non-parallel vectors in $\mathbb R^3$ span a plane through the origin. Notice the origin is always included: set every weight to $0$ and you haven't moved.

The key equivalence: $\vec b$ is in $\operatorname{Span}\{\vec a_1, \dots, \vec a_n\}$ exactly when the system with augmented matrix $[\vec a_1\;\cdots\;\vec a_n \mid \vec b]$ is consistent. Think about why. Being in the span means some weights $x_1, \dots, x_n$ build $\vec b$, and those weights are exactly a solution of that system. So everything you learned about row reduction carries straight over: to test whether a vector can be reached, row reduce and look for a $0 = b$ row.

A common mistake. It's tempting to think two vectors in $\mathbb R^3$ can reach any point in space, because two vectors were enough to fill the plane $\mathbb R^2$. They can't: two non-parallel vectors in $\mathbb R^3$ only span a plane through the origin, and every point off that plane is out of reach. To fill all of $\mathbb R^3$ you need at least three vectors.

There's an animation further down that shows two thrusters' pushes adding up to a target.

Mixing problems are vector equations: which combination of available alloys (or nutrients, or paint colors) produces a target composition?