Foundations — Dimension — basis cardinality

Before you can trust the sentence "every basis has the same number of vectors", you must know what a vector, a vector space, a linear combination, independence, spanning, a basis, and cardinality actually are — as pictures, not just words. We build them in the order they depend on each other.

1. Vector — an arrow you can stretch and add

We write vectors with lowercase letters like $v$, $w$. In the plane ${\mathbb{R}}^2$ ("all pairs of real numbers") a vector is a pair $(x,y)$: walk $x$ steps right, then $y$ steps up, plant the arrow tip there.

Why the topic needs it. The whole theorem counts vectors in a basis. If you don't picture the object being counted, nothing that follows means anything.

2. Scaling and adding — the two allowed moves

Why the topic needs it. The next concept, "linear combination", is literally just these two moves used together.

3. Linear combination — mix your ingredients

The symbol $\sum$ ("sigma") is shorthand for "add all of these up": ${\sum }_{i=1}^n{a}_i{w}_i$ means exactly the sum above. The subscript $i$ is just a counter running from $1$ to $n$.

Why the topic needs it. The exchange lemma's very first step writes $v_1=a_1{w}_1+\cdots +a_n{w}_n$. That line is a linear combination. Every proof step rearranges one.

4. Spanning — do your ingredients reach everywhere?

The symbol $\in$ means "is a member of": $v\in V$ reads "$v$ is a vector in the space $V$". The symbol $\subseteq$ means "is a subset of": $S\subseteq V$ reads "$S$ is a collection of vectors all living inside $V$".

Why the topic needs it. Steinitz says "independent $\le$ spanning". The spanning set provides the slots that get filled. No notion of spanning, no lemma.

5. Linear independence — no wasted ingredients

The symbol $⟹$ means "forces" / "implies": the left statement being true forces the right statement to be true.

Why the topic needs it. In the lemma's key step, independence is exactly what guarantees "some coefficient on a leftover $w$ is nonzero", so a swap is always possible. See Linear Independence.

6. Basis — just enough, no more, no less

Why the topic needs it. The theorem measures the size of a basis. A basis plays both roles — independent AND spanning — which is why the exchange lemma can be applied to it in both directions. See Basis of a Vector Space and Spanning Sets.

7. Cardinality — the count itself

Why the topic needs it. The entire theorem is one equation about cardinalities: $|B_1|=|B_2|$. The number they share is the dimension $\dim V$.

8. Dimension — the pinned-down number

How the pieces feed the theorem

Equipment checklist

Test yourself — cover the right side.

Connections

• Linear Independence — the "no redundancy" property built in §5.
• Spanning Sets — the "reaches everything" property built in §4.
• Basis of a Vector Space — the both-at-once object of §6.
• Steinitz Exchange Lemma — the engine that turns these foundations into the count theorem.
• Rank-Nullity Theorem — dimension counting applied to linear maps.
• Coordinates and Change of Basis — what changes (and what doesn't) when you switch basis.