Rongzhou: Created page with "== Vectors == === Mathematical definition === A set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is '''linearly independent''' if $$ c_1 \vec{v}_1 + c_2 \vec{v}_2 + \cdots + c_n \vec{v}_n = \vec{0} $$ has no solution besides $$ c_1, c_2, \cdots, c_n = 0. $$ === Equivalent definition === Equivalently, it can be stated that if $$ \forall i,j \in \{1,n\} \vec v_i \neq \sum \lambda\vec v_j \\ \text{(no vector in the set can be written as a linear combinat..."

2026-10-12T08:42:19Z

Created page with "== Vectors == === Mathematical definition === A set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is '''linearly independent''' if $$ c_1 \vec{v}_1 + c_2 \vec{v}_2 + \cdots + c_n \vec{v}_n = \vec{0} $$ has no solution besides $$ c_1, c_2, \cdots, c_n = 0. $$ === Equivalent definition === Equivalently, it can be stated that if $$ \forall i,j \in \{1,n\} \vec v_i \neq \sum \lambda\vec v_j \\ \text{(no vector in the set can be written as a linear combinat..."

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== Vectors ==

=== Mathematical definition ===

A set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is '''linearly independent''' if

$$
c_1 \vec{v}_1 + c_2 \vec{v}_2 + \cdots + c_n \vec{v}_n = \vec{0}
$$

has no solution besides

$$
c_1, c_2, \cdots, c_n = 0.
$$

=== Equivalent definition ===

Equivalently, it can be stated that if

$$
\forall i,j \in \{1,n\} \vec v_i \neq \sum \lambda\vec v_j \\
\text{(no vector in the set can be written as a linear combination of some others)}
$$

then set of vectors $\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is '''linearly independent'''

Linearly independent - Revision history