TITLE: Most harmful heuristic?
QUESTION [167 upvotes]: What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?

REPLY [6 votes]: The "size" of a finite-dimensional vector space is proportional to its dimension.

In fact, the "size" of a finite-dimensional vector space is almost always better thought of as being exponential in its dimension. This is easiest to see for (finite-dimensional) vector spaces over finite fields, which have finite cardinality. But it's a better heuristic even for vector spaces over infinite fields.
Internalizing the correct intuition makes it clear why forming the (algebraic) direct product of two vector spaces causes their dimensions to add, and not to multiply as you might naively expect based on the fact that taking the direct product of groups multiplies their orders.
Another confusing point to which this misconception leads regards the advantage that quantum computers give over classical ones. The difference is sometimes stated as "quantum computers have a state space that's exponentially large in the number of qubits," but this is highly misleading, because classical computers also have a state space that's exponentially large in the number of bits. The better intuition is: since quantum computers have a state space whose dimension is exponentially large in the number of qubits, the state space itself is actually doubly exponential in the number of qubits, while the state space of a classical computer is only singly exponential in the number of bits.
The reason why this misconception is so widespread is that early courses in linear algebra almost always begin with vectors spaces over infinite fields (usually $\mathbb{R}$ or $\mathbb{C}$), which have infinite cardinality, so the dimension is the only finite number available. This practice leads to misleading intuition for general vector spaces.