Independence, Span, Basis & Dimension

Here is the word that crossed the border from probability — independence — now meaning something about arrows. Pick a handful of them. How few can you keep and still reach every point of a space by adding and scaling? An arrow that points in a genuinely new direction earns its place; one that is secretly a stretch of the others is dead weight. That single question opens four tightly linked ideas — independence, span, basis, dimension — and GATE tests them as a cluster.

Independence — no redundant directions

Vectors v₁, …, vₙ are linearly independent when no one of them is a combination of the others. The crisp algebraic test: the only way to make

c₁v₁ + c₂v₂ + … + cₙvₙ = 0

is to take every coefficient cᵢ = 0. If some non-zero choice of coefficients also gives zero, the vectors are dependent — at least one is redundant, addable from the rest.

Drag the two arrows below. Point them different ways and every combination fills out the plane — independence in action. Now lay one on top of (or opposite) the other and the combinations collapse onto a single line — the hallmark of dependence.

Span, basis, dimension

The span of a set of vectors is everything you can build from them by adding and scaling — all their linear combinations. Two independent vectors in R² span the whole plane; one non-zero vector spans only a line.

A basis of a space is a set that is both independent and spanning — just enough arrows to reach everything, with none redundant. The dimension is simply the number of vectors in a basis (every basis of a given space has the same count). So R² has dimension 2, R³ has dimension 3. And a space has many valid bases: any n independent vectors that span an n-dimensional space will do — there is nothing special about the “standard” one.

Orthonormal — perpendicular and unit length

A set is orthonormal when its vectors are mutually perpendicular (every pair has dot product 0) and each has length 1. The standard basis (1,0), (0,1) is orthonormal. A fact GATE tests: an orthonormal set is automatically independent — perpendicular directions can never be combinations of one another. But the converse is weaker: independent vectors need not be perpendicular.

A worked example — independent or dependent?

(1) Are (1, 0) and (1, 1) independent, and do they form a basis of R²? (2) Are (1, 2) and (2, 4) independent?

For two vectors in R², the quickest test is the determinant of the matrix they form — non-zero means independent:

det [1 1]  = 1·1 − 0·1 = 1  ≠ 0   →  (1,0),(1,1) independent → a basis of R²
    [0 1]

det [1 2]  = 1·4 − 2·2 = 0        →  (1,2),(2,4) dependent
    [2 4]

The first pair are independent, and two independent vectors in the 2-dimensional R² automatically span it, so they are a basis — a different basis from the standard (1,0), (0,1). The second pair are dependent because (2, 4) = 2·(1, 2): the second adds no new direction, so they span only a line.

A question to carry forward

For a clutch of arrows we just asked “how many point in genuinely new directions?”. A matrix is exactly a stack of column arrows — so the same question applies to it. Here is the thread onward: how many independent directions does a matrix hold, what do we call that number, and what does it instantly tell us about the equation Ax = b?

In one breath

• Independent ⇔ c₁v₁+…+cₙvₙ = 0 forces all cᵢ = 0 (no vector is a combination of the others); a non-zero solution means dependent (a redundant direction).
• Span = all linear combinations; basis = independent and spanning; dimension = number of basis vectors (every basis has the same count; Rⁿ → n).
• A space has many bases — any n independent vectors that span.
• Orthonormal (perpendicular + unit length) ⇒ automatically independent, but independent ⇏ orthogonal ((1,0),(1,1) are independent, not perpendicular).
• Fast check for n vectors in Rⁿ: form a matrix — det ≠ 0 ⇔ independent (a basis). Any n+1 vectors in Rⁿ are always dependent.

Practice