Linear_combination_Linearly_dependent_Linearly_independent ========================================================= Linear combination Suppose you have n number of vectors : $$$x = \{x_{1}, x_{2},...,x_{N}\}$$$ Suppose you have n number of scalars : $$$a = \{a_{1}, a_{2},...,a_{N}\}$$$ You can consider this notation: $$$y = a_{1}x_{1}+a_{2}x_{2}+...+a_{N}x_{N}$$$ y which is created in this way should be also vector. With this notation, you can say "vector y" is "linear combination" of "vector x" or "vector y" is created based on "linear combination" of "vector x" ========================================================= 2 characteristics (linearly dependent, linearly independent) of vector created by linear combination -------------------------------------------------- 1. Linearly dependent There is vector set $$$x=\{x_{1},x_{2},...,x_{N}\}$$$ If you can find scalar coefficient set $$$\{a_{1},a_{2},...,a_{N}\}$$$ which satisfies following equation, $$$a_{1}x_{1}+a_{2}x_{2}+...+a_{N}x_{N}=\sum\limits_{i=1}^{N}a_{i}x_{i}=0$$$ you can say "vector set" $$$x=\{x_{1},x_{2},...,x_{N}\}$$$ is "linearly dependent" Trivial solution is all scalar coefficients are 0. But you're not thinking of that trivial solution. You can see example of linearly dependent case like this: $$$\begin{bmatrix} 0\\0\\0 \end{bmatrix} = 1\times \begin{bmatrix} 2\\2\\4 \end{bmatrix} -2\times \begin{bmatrix} 2\\2\\3 \end{bmatrix} + 2\times \begin{bmatrix} 1\\1\\1 \end{bmatrix}$$$ So, you can say above 3 vectors $$$\begin{bmatrix} 2\\2\\4 \end{bmatrix}$$$, $$$\begin{bmatrix} 2\\2\\3 \end{bmatrix}$$$, $$$\begin{bmatrix} 1\\1\\1 \end{bmatrix}$$$ are linearly dependent because there exists set of coefficient $$$\{1,-2,2\}$$$ Or, you can say "definition of linearly dependent" like this : $$$a_{1}x_{1}+a_{2}x_{2}+...+a_{N-1}x_{N-1}=-a_{N}x_{N}$$$ Elements $$$(x_{1},x_{2},...)$$$ can make other elements $$$(x_{N},...)$$$, which means they're related to each other, so you can use term "linearly dependent" ========================================================= 2. Linearly independent It's more simple concept than linearly dependent. Linearly independent indicates case that when solution satisfies following equation: $$$a_{1}x_{1}+a_{2}x_{2}+...+a_{N}x_{N}=\sum\limits_{i=1}^{N}a_{i}x_{i}=0$$$ Set satisfying above equation is only one set where all scalar coefficients are 0, you can say "vectors" out of vector set x are linearly independent to each other. Example : $$$\begin{bmatrix} 0\\0\\0 \end{bmatrix} = 0\times \begin{bmatrix} 1\\0\\0 \end{bmatrix} +0\times \begin{bmatrix} 0\\1\\0 \end{bmatrix} + 0\times \begin{bmatrix} 0\\0\\1 \end{bmatrix}$$$ ========================================================= 3 vectors which are linearly independent can be plotted on space like below: Vectors located on each axis are linearly independent. Basis vectors are basically linearly independent.