08. Linear Independence
Dated: 12-03-2025
Definition
An indexed set1 of vectors2 in \(\mathbb R^n\) is said to be linearly independent if \(x_1 \vec v_1 + x_2 \vec v_2 + \cdots + x_p \vec v_p = 0\) has only trivial solution.3
Linear Independence of Matrix Column
The columns of a matrix \(A\) are linearly independent if and only if the equation \(A \vec x = \vec 0\) has only the trivial solution.3
Set of One Vector
The set1 \(\{\vec v\}\) is linearly independent if \(\vec v\) is not a null vector.2 \(x_1 \vec v = \vec 0\) only trivial solution3 when \(\vec v \ne \vec 0\).
The equation \(x_1 \vec 0 = \vec 0\) has many non trivial solutions that is why \(\vec v \ne \vec 0\).
Theorem
An indexed set \(S = \{\vec {v_1}, \vec{v_2}, \ldots, \vec{v_n}\}\) is linearly dependent if at least one vector2 \(\vec {v_i} \in S\) is a linear combination4 of the others.
There will be vectors2 in \(S\) which are NOT linear combinations4 of the others, despite \(S\) being linearly dependent.
Theorem
A set1 \(S = \{\vec {v_1}, \vec {v_2}, \ldots, \vec {v_p}\} \in \mathbb R^n\) is linearly dependent if \(p > n\).
Theorem
A set1 \(S = \{\vec {v_1}, \vec {v_2}, \ldots, \vec {v_p}\} \in \mathbb R^n\) is linearly dependent if \(\vec 0 \in S\).
References
Read more about notations and symbols.