independence of vectors in R^n space

1. The Space Rⁿ

Rⁿ is the set of all ordered n-tuples of real numbers, written as row vectors (x₁, x₂, …, xₙ) or column vectors. With the usual addition and scalar multiplication it is a vector space, and its dimension is n.

R^n = {(x_1, x_2, ..., x_n) : x_i ∈ R}, dim R^n = n

The standard (canonical) basis of Rⁿ is the set of unit vectors

e_1 = (1,0,...,0), e_2 = (0,1,...,0), ..., e_n = (0,0,...,1)

In R² this is i, j = (1,0), (0,1); in R³ it is i, j, k = (1,0,0), (0,1,0), (0,0,1). Every vector of Rⁿ is a unique combination of these: (a, b, c) = a i + b j + c k.

2. Four Definitions That Fit Together

Term	Meaning
Span	The set of all linear combinations of the given vectors
Spanning set	A set whose span is the whole space (every vector can be built from them)
Linearly independent set	c₁v₁ + … + cₙvₙ = 0 only when all cᵢ = 0 — no vector is redundant
Basis	A set that is both linearly independent and spanning — the minimum number of vectors that reach every point of the space
Dimension	The number of vectors in any basis of the space (well-defined: all bases of a space have the same size)

Mnemonics: independent = no redundancy, spanning = no gaps, basis = neither redundancy nor gaps.

3. The Independence Theorems for Rⁿ

Maximum size. In Rⁿ, at most n vectors can be linearly independent. Any set of n + 1 or more vectors in Rⁿ is dependent.
Minimum size for spanning. Fewer than n vectors can never span Rⁿ (they span at most a subspace of smaller dimension).
Exactly n is the sweet spot. Any set of exactly n linearly independent vectors in Rⁿ is automatically a basis of Rⁿ (it necessarily spans). Equivalently, n vectors in Rⁿ span Rⁿ ⇔ they are independent ⇔ the n × n matrix of the vectors is non-singular (det ≠ 0).
Dimension of a subspace. If W is a subspace of Rⁿ then 0 ≤ dim W ≤ n; dim W = n only when W = Rⁿ.
Rank link. For vectors written as rows of A: the number of independent vectors = ρ(A); the row space of A has dimension ρ(A), and so does the column space — this is the meaning of "row rank = column rank".

   Given m vectors in R^n:
   ┌── m > n  ──────────────────────────► DEPENDENT (by theorem 1, no work needed)
   └── m ≤ n
        └── arrange them as rows of A (m × n)
             └── reduce to echelon form
                  ├── number of non-zero rows = m  ──► INDEPENDENT
                  └── number of non-zero rows < m  ──► DEPENDENT
   Special case m = n : write the square matrix and test det ≠ 0.

8. Independence, Basis and Rank — One Summary Table

Statement	Equivalent formulation
n vectors in Rⁿ are independent	The n × n matrix is non-singular (det ≠ 0); rank = n
They form a basis of Rⁿ	They are independent and span Rⁿ (for exactly n vectors, either property implies the other)
The set is dependent	Rank of the matrix of vectors < number of vectors
Dimension of the span	= rank of the matrix of vectors
Rⁿ has dimension n	Standard basis e₁, …, eₙ has n vectors

9. Exam Points

Define linear independence, span, basis and dimension for Rⁿ — 5 marks, almost certain.
Show that n linearly independent vectors in Rⁿ form a basis of Rⁿ — 5-7 marks, very frequent.
Prove that any set of more than n vectors in Rⁿ is linearly dependent — 5 marks.
Test whether a given set of 3 vectors is a basis of R³ — 5 marks.
Express a given vector as a linear combination of a given basis (find coordinates) — 5-7 marks, frequent.
Reduce a dependent spanning set to a basis — 5 marks.
State the standard basis of R³ and the dimension of Rⁿ — 3 marks.

independence of vectors in R^n space