Basis and Dimension

The Coordinate Skeleton

A basis for a vector space $V$ is a set of vectors that is linearly independent (no redundancy) and spanning (reaches every vector in $V$ ). It is a minimal spanning set: just enough vectors to build everything.

Concept	Definition	Rule
Basis	Independent + Spanning	Minimal spanning set
Standard Basis	$\mathbb{R}^n$ unit vectors	$e_1 = (1,0,\dots)$
Dimension $\dim V$	Size of any basis	Fundamental invariant

Every basis of a given space has the exact same number of vectors. That number is the dimension $\dim V$ (e.g., $\mathbb{R}^2$ is 2, $\mathbb{R}^3$ is 3).

Two vectors reach a marked target; drop one and the target goes dark

Coordinates & Shortcuts

Because a basis spans, every vector is a combination of basis vectors; because it is independent, that combination is unique. This gives every vector a unique list of coordinates.

Test in $\dim n$ Space	Result	Pitfall
$n$ independent vectors	Automatically a basis	Don't assume spanning
$n$ spanning vectors	Automatically a basis	Don't assume independence

Common pitfall: Thinking any spanning set or independent set alone is a basis. A basis requires both properties, and always contains exactly $\dim V$ vectors.

The Coordinate Skeleton

Coordinates & Shortcuts

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Linear Algebra