Linear Dependence and Independence of Rows

Let $α_{1}, α_{2}, \dots, α_{p}$ be the rows of a matrix $A_{m \times n}$ (they can be added and multiplied by a number just like ordinary row vectors).

Definition. The row $α_{p}$ is said to be linearly expressed through the rows $α_{1}, \dots, α_{p - 1}$ (to be their linear combination) if there exist numbers $c_{1}, \dots, c_{p - 1} \in R$ such that

α_{p} = c_{1} α_{1} + c_{2} α_{2} + \dots + c_{p - 1} α_{p - 1} .

Definition. The rows $α_{1}, α_{2}, \dots, α_{p}$ are called linearly dependent if there exist coefficients $k_{1}, k_{2}, \dots, k_{p}$ , not all equal to zero (i.e. $k_{1}^{2} + k_{2}^{2} + \dots + k_{p}^{2} \neq = 0$ ), such that

k_{1} α_{1} + k_{2} α_{2} + \dots + k_{p} α_{p} = 0, (1)

where $0$ is the zero row. If equality (1) is possible only when $k_{1} = k_{2} = \dots = k_{p} = 0$ , the rows are called linearly independent.

Equivalence of the Two Notions

Theorem. The rows are linearly dependent if and only if at least one of them is linearly expressed through the rest.

Proof. ( $\Leftarrow$ ) Suppose $α_{p} = c_{1} α_{1} + \dots + c_{p - 1} α_{p - 1}$ . Move everything to one side:

c_{1} α_{1} + \dots + c_{p - 1} α_{p - 1} - α_{p} = 0.

This is a relation of form (1) with coefficients $k_{1} = c_{1}, \dots, k_{p - 1} = c_{p - 1}, k_{p} = - 1 \neq = 0$ — not all zero. So the rows are linearly dependent.

( $\Rightarrow$ ) Suppose (1) holds and, say, $k_{p} \neq = 0$ . Then

α_{p} = - \frac{k _{1}}{k _{p}} α_{1} - \frac{k _{2}}{k _{p}} α_{2} - \dots - \frac{k _{p - 1}}{k _{p}} α_{p - 1},

that is, $α_{p}$ is expressed through the rest.

Basic Properties

A system containing the zero row is linearly dependent (take coefficient $1$ for the zero row and $0$ for the rest).
A system containing two equal or proportional rows is linearly dependent.
If part of the rows is linearly dependent, then the whole system is linearly dependent.

Relation to Rank

Theorem. If $r (A) = k$ , then the matrix $A$ has $k$ linearly independent rows through which all the remaining rows are linearly expressed.

In other words, the rank of a matrix equals the maximum number of its linearly independent rows (and, by the invariance of rank under transposition, of its columns). In particular, for a square matrix of order $n$ , the rows are linearly independent $⟺ r (A) = n ⟺ det A \neq = 0$ (see Rank of a Matrix and Properties of Determinants).

Example

Let's determine whether the rows (vectors)

α_{1} = (2, - 1, 1), α_{2} = (1, 2, - 1), α_{3} = (3, - 4, 3)

are linearly dependent. Form a linear combination and set it equal to the zero row: $λ_{1} α_{1} + λ_{2} α_{2} + λ_{3} α_{3} = 0$ . Coordinate-wise, this is the homogeneous system

⎩ ⎨ ⎧ 2 λ_{1} + λ_{2} + 3 λ_{3} = 0, - λ_{1} + 2 λ_{2} - 4 λ_{3} = 0, λ_{1} - λ_{2} + 3 λ_{3} = 0.

Gaussian elimination reduces it to

100010 2 - 1 0 ∣ ∣ ∣ 000 \Rightarrow λ_{1} λ_{2} λ_{3} = C - 2 11, C \in R .

There are nonzero solutions, so the rows are linearly dependent. Taking $C = 1$ , we obtain the relation $- 2 α_{1} + α_{2} + α_{3} = 0$ , that is, $α_{3} = 2 α_{1} - α_{2}$ .

The same follows from the determinant of the matrix formed from these rows:

213 - 1 2 - 4 1 - 1 3 = 0,

and the determinant being zero is exactly what linear dependence of the rows means (the rank is less than $3$ ).

Equivalence of the Two Notions

Basic Properties

Relation to Rank

Example

Related works