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A basis of vector space V is a linearly independent spanning set. Every basis has the same cardinality, called the dimension of V. Dimension is the minimum number of vectors needed to span V and the maximum size of a linearly independent set. Coordinates relative to a basis are unique.
A basis is the most economical way to describe a vector space: it is a set of vectors that is both large enough to reach everywhere in the space (spanning) and small enough to have no redundancy (linear independence). Think of it like a coordinate system — the standard basis vectors e₁ = (1,0) and e₂ = (0,1) in ℝ² let you describe any vector as a unique combination, like (3,−2) = 3e₁ + (−2)e₂. If you added a third vector like (1,1), you would have redundancy; if you removed one, you could no longer reach all of ℝ².
The central theorem is that every basis for a given vector space has the same number of vectors. This common count is the dimension of the space. It does not matter which basis you pick — the standard basis, a rotated basis, an unusual-looking basis — they all have the same size. This is what makes dimension a well-defined property of the space itself, not of any particular basis. ℝ³ has dimension 3, the space of polynomials of degree ≤ 2 has dimension 3 (basis: {1, x, x²}), and the zero vector space has dimension 0.
Dimension has two equivalent characterizations that are worth internalizing: it is the *minimum* number of vectors needed to span the space, and the *maximum* number of vectors that can be linearly independent. These are dual perspectives on the same fact. Any spanning set with exactly n vectors must be independent (hence a basis). Any independent set with exactly n vectors must span (hence a basis). So checking either condition, if the count is right, automatically gives you the other.
The uniqueness of coordinates is the payoff. Once you fix a basis {v₁, …, vₙ}, every vector w in the space can be written as w = c₁v₁ + ⋯ + cₙvₙ in exactly one way. Those scalars (c₁, …, cₙ) are the coordinates of w in that basis. This is why linear independence matters so deeply — if the basis vectors were linearly dependent, coordinates would not be unique, and the whole coordinate system would break down.