MA25C02| Linear Algebra | Vector Space| 🧮 Linear Independence & Dependence | Definitions, Properties & Problems — Linear Algebra Notes

Linear Algebra  ·  Vector Spaces

Linear Independence & Linear Dependence

Understanding when vectors truly span new directions — and when they don't.

Vector Spaces Basis Span Dimension

▶ Tutorial Video: Linear Independence and Linear Dependence

Section 1

Linear Independence

Definition

A subset $S$ of a vector space $V$ is called Linearly Independent if, for any finite collection of distinct vectors $u_1, u_2, \dots, u_n \in S$, the equation

$$a_1 u_1 + a_2 u_2 + \cdots + a_n u_n = 0$$

holds only when $a_1 = a_2 = \cdots = a_n = 0$. That is, the trivial combination is the only way to produce the zero vector.

Note

For any vectors $u_1, u_2, \dots, u_n$, we always have $a_1 u_1 + a_2 u_2 + \cdots + a_n u_n = 0$ when $a_1 = a_2 = \cdots = a_n = 0$. This is called the trivial solution. Linear independence demands it be the only solution.

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Section 2

Linear Dependence

Definition

A subset $S$ that is not linearly independent is called Linearly Dependent. There exist scalars $a_1, a_2, \dots, a_n$, not all zero, such that

$$a_1 u_1 + a_2 u_2 + \cdots + a_n u_n = 0$$

At least one vector can be expressed as a linear combination of the others — it is "redundant" and adds no new direction.

Elementary Facts

• The empty set is linearly independent — a linearly dependent set must be non-empty by definition.
• A set $\{u\}$ containing a single non-zero vector is always linearly independent. If $au = 0$ for non-zero $a$, then $u = a^{-1} \cdot 0 = 0$, a contradiction.
• A set is linearly independent if and only if the only linear combination of its vectors equal to $0$ is the trivial one.

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Section 3

Intuitive Meaning

Linearly Independent

Each vector adds a new direction that cannot be reached by the others. No vector in the set is a linear combination of the rest.

Linearly Dependent

At least one vector is redundant — it can be written as a combination of the others. The vectors live in a smaller subspace than their count suggests.

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Section 4

Key Properties

01

Any set containing the zero vector is linearly dependent.

02

A set of two vectors is linearly dependent if and only if one is a scalar multiple of the other.

03

In $\mathbb{R}^n$, the maximum size of a linearly independent set is $n$.

04

If a set is linearly independent, every subset is also linearly independent.

05

If a set is linearly dependent, any superset (adding more vectors) is also linearly dependent.

06

$n$ vectors in $\mathbb{R}^n$ are linearly independent if and only if $\det(A) \neq 0$, where $A$ is the matrix formed with these vectors as columns.

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Section 5

Relationship with Span and Basis

Key Insight

Linear independence, span, and basis form the structural backbone of every vector space.

• A basis is a set that is both linearly independent and spans the space — the most efficient description of the space.
• Removing dependent vectors from a set does not change its span — those vectors contributed nothing new.
• The dimension $\dim(V)$ equals the size of any basis: the maximum size of a linearly independent set, and the minimum size of a spanning set.

$$\underbrace{\text{Basis}}_{\text{Independent} + \text{Spanning}} \;\Longrightarrow\; \dim(V) = \text{size of any basis}$$

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Section 6

Testing for Linear Independence

Step-by-Step Method: Row Reduction

Given vectors $v_1, \dots, v_k$ in $\mathbb{R}^n$, follow these steps:

1. Form matrix $A$ whose columns are the vectors $v_1, \dots, v_k$.
2. Row reduce $A$ to row echelon form.
3. If every column has a pivot → vectors are linearly independent.
4. If any column lacks a pivot → vectors are linearly dependent.

Alternative — Determinant Method For a square matrix ($k = n$), compute $\det(A)$. If $\det(A) \neq 0$ → independent; if $\det(A) = 0$ → dependent.

Linear Algebra Notes  ·  Vector Spaces Series