Dimension of a subset of $\mathbb R^n$

Question

I'm currently reading Linear Algebra and Its Applications by David C. Lay. Chapter 8 section 1 defines the dimension of a set S (in this context $S$ is understood to be a subset of $\mathbb R^n$) as the dimension of the smallest flat containing $S$.

I want to know why this definition is at all useful because wouldn't it imply that a straight line has dimension $1$, but a curve in $\mathbb R^3$ that twists and turns has dimension $3$?

Intuitively, I would say that both of those objects are $1$ dimensional, so why does he define dimension like this?

Answer 1

Although I am unfamiliar with the definition, It makes sense to define dimension like this because it would take at least 3 vectors to span the curve. In other contexts, you would be right, the curve can be said to be a 1 dimensional manifold, that is, each point on the curve is locally "homeomorphic"(i.e. looks like) a line.