Linear dependence

1. Linear Combination

A vector v is a linear combination of the vectors v₁, v₂, …, vₙ if scalars c₁, c₂, …, cₙ exist such that

v = c_1 v_1 + c_2 v_2 + ... + c_n v_n

The set of all such combinations is the span of {v₁, …, vₙ}.

2. Definition of Dependence and Independence

The vectors v₁, v₂, …, vₙ are linearly dependent if there exist scalars c₁, …, cₙ, not all zero, such that

c_1 v_1 + c_2 v_2 + ... + c_n v_n = 0

They are linearly independent if this equation has only the trivial solution c₁ = c₂ = … = cₙ = 0.

In words: a dependent set contains redundant vectors — at least one of them can be written as a combination of the others. An independent set contains no redundancy; every vector adds a new direction.

3. Geometric Meaning

Number of vectors	In R²	In R³
2 dependent	both on the same line (parallel)	same line
2 independent	different directions	different directions
3 dependent	—	all lie in one plane through the origin (coplanar)
3 independent	—	span the whole space R³

Also geometrically obvious: a set containing the zero vector is always dependent (take its coefficient as 1 and the rest 0).

4. Three Tests to Decide Dependence

Test	Rule
1. Equation test (definition)	Solve c₁v₁ + … + cₙvₙ = 0. Non-trivial solution ⇒ dependent; only the trivial solution ⇒ independent.
2. Rank test	Arrange the vectors as the rows of a matrix A. Then ρ(A) = n ⇒ independent, ρ(A) < n ⇒ dependent, where n = number of vectors.
3. Determinant test	If there are exactly n vectors in Rⁿ, form the n × n matrix. det ≠ 0 ⇒ independent; det = 0 ⇒ dependent.

5. Standard Theorems

Any set containing the zero vector is linearly dependent.
Any set of more than n vectors in Rⁿ is necessarily dependent.
A subset of a linearly independent set is linearly independent; a superset of a dependent set is dependent.
If a set is dependent, then at least one vector is a linear combination of the others (but not necessarily every one).
n linearly independent vectors in Rⁿ form a basis of Rⁿ (see the next topic).
If {v₁, …, vₙ} is independent and w is not in its span, then {v₁, …, vₙ, w} is also independent.

  Check :  v1 = (1, 2, 3) ,  v2 = (3, -2, 1) ,  v3 = (1, -6, -5)

  Determinant test (3 vectors in R^3):
       | 1   2   3 |
  det  | 3  -2   1 | = 1[(-2)(-5) - (1)(-6)] - 2[(3)(-5) - (1)(1)] + 3[(3)(-6) - (-2)(1)]
       | 1  -6  -5 |
                     = 1(10 + 6) - 2(-15 - 1) + 3(-18 + 2)
                     = 16 + 32 - 48 = 0     ⇒  D E P E N D E N T

  The actual relation (find it by solving the homogeneous system):
     2 v1 - v2 + v3 = (2 - 3 + 1,  4 + 2 - 6,  6 - 1 - 5) = (0, 0, 0)   ✓
  so  c1 = 2 , c2 = -1 , c3 = 1  is a non-trivial solution, confirming dependence.

9. Dependence and Rank — the Connection to Remember

For n vectors in Rᵐ, dependence is exactly a rank deficiency: if the vectors are written as rows of A, then

independent ⇔ ρ(A) = n, dependent ⇔ ρ(A) < n

The number of independent vectors in the set equals the rank, and the remaining (n − ρ) vectors are redundant combinations of them. This single idea covers the equation test, the determinant test (for square matrices det = 0 means rank < n) and the "more than n vectors in Rⁿ" rule.

10. Exam Points

Define linear dependence and independence of vectors, with the equation c₁v₁ + … + cₙvₙ = 0 — 5 marks, almost certain.
Test whether a given set of vectors in R³ is dependent (determinant or rank method) — 5-7 marks, very frequent.
Find the value(s) of k for which a given set of vectors is linearly dependent — 5-7 marks, very frequent.
Show that any set containing the zero vector is dependent — 3 marks.
Show that more than n vectors in Rⁿ are always dependent — 3-5 marks.
Express one of three dependent vectors as a linear combination of the other two — 5 marks.
State the geometric meaning of dependence for 2 and 3 vectors — 3 marks.

Linear dependence