Linear combinations

Take $u = (1, 0)$ and $v = (0, 1)$. Scale each by anything, add the results:

$$a,u + b,v = a(1,0) + b(0,1) = (a, b)$$

Any expression of that shape — scale, then add — is a linear combination. Watch the visual sweep through values of $a$ and $b$, tracing where the tip lands.

The reachable set

Look at the answer: $(a, b)$. Choosing $a$ and $b$ freely produces any pair of numbers, so every point in the plane is reachable. Nothing is out of range.

Now change one thing. Keep $u = (1,0)$, but make $v = (2, 0)$:

$$a(1,0) + b(2,0) = (a + 2b,; 0)$$

The second coordinate is stuck at zero, whatever you choose. You can reach $(7, 0)$ and $(-3, 0)$, but $(1, 1)$ is impossible. Two vectors, and they only cover a single line.

Why the second case collapsed

$v = (2,0)$ was already a scaled copy of $u$. It offered no direction $u$ did not already provide, so it bought nothing.

That is the first genuinely important idea here: what matters is not how many vectors you have, but how many independent directions they give you. Two vectors along one line are worth exactly one vector.

The recipe reading

Return to $a,u + b,v = (a,b)$ and read it backwards. The coordinates $(a,b)$ are the amounts of $u$ and $v$ in the recipe.

Coordinates were never really points in space. They are always amounts of chosen ingredients. That sounds like a triviality when the ingredients are $(1,0)$ and $(0,1)$. It becomes the key to the entire second half of this path once the ingredients change.

Two questions follow, and they define the next two steps: what exactly does a set of vectors reach, and what makes a good set of ingredients?