Determining Linearly Dependent Vectors I am learning about Linear dependent vector from here
But I am unable to grasp the following equation:

If no such scalars exist, then the vectors are to be linearly independent.
$$c_1\begin{bmatrix}x_{11}\\x_{21}\\\vdots\\x_{n1}\\ \end{bmatrix}+c_2\begin{bmatrix}x_{12}\\x_{22}\\\vdots\\x_{n2}\\ \end{bmatrix}+\cdots+c_n\begin{bmatrix}x_{1n}\\x_{2n}\\\vdots\\x_{nn}\\ \end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\\ \end{bmatrix}\\
\begin{bmatrix}x_{11}&x_{12}&\cdots&x_{1n}\\x_{21}&x_{22}&\cdots&x_{2n}\\ \vdots&\vdots&\ddots&\vdots\\x_{n1}&x_{n2}&\cdots&x_{nn}&\\ \end{bmatrix}\begin{bmatrix}c_1\\c_2\\\vdots\\c_n\end{bmatrix}=\begin{bmatrix}0\\0\\\vdots\\0\end{bmatrix}$$
   In order for this matrix equation to have a nontrivial solution, the determinant must be $0$

How the first equation is reduced to the second one?
 A: I hope this small example can help you to understand.
$$\begin{bmatrix} x_{11}  & x_{12} \\ x_{21} & x_{22} \end{bmatrix}\begin{bmatrix} c_{1}   \\ c_{2}\end{bmatrix}=\begin{bmatrix} c_{1}x_{11}+c_2x_{12}   \\ c_{1}x_{21}+c_2x_{22}  \end{bmatrix}=c_1\begin{bmatrix} x_{11} \\ x_{21}\end{bmatrix}+ c_2\begin{bmatrix} x_{21} \\ x_{22}\end{bmatrix}$$
Notice that $c_1$ is only multiplied to entries in the first column and $c_2$ is only multiplied to the entries in the second column.
A: This results from the definition of scalar  multiplication and addition of matrices:
\begin{align}
c_1\begin{bmatrix}
x_{11}\\x_{21}\\\vdots\\x_{n1}
\end{bmatrix}+c_2\begin{bmatrix}
x_{12}\\x_{22}\\\vdots\\x_{n2}
\end{bmatrix}+\dots +c_n\begin{bmatrix}
x_{1n}\\x_{2n}\\\vdots\\x_{nn}
\end{bmatrix} &= \begin{bmatrix}
c_1x_{11}\\c_1x_{21}\\\vdots\\c_1x_{n1}
\end{bmatrix}+\begin{bmatrix}
c_2x_{12}\\c_2x_{22}\\\vdots\\c_2x_{n2}
\end{bmatrix}+\dots +\begin{bmatrix}
c_nx_{1n}\\c_nx_{2n}\\\vdots\\c_nx_{nn}
\end{bmatrix}\\[1ex]
&= \begin{bmatrix}
c_1x_{11}+c_2x_{12}+\dots+c_nx_{1n}\\c_1x_{21}+c_2x_{22}+\dots+c_nx_{2n}\\\dots\dots\dots\dots\dots\dots\dots\dots\\c_1x_{n1}+c_2x_{n2}+\dots+c_nx_{nn}
\end{bmatrix} 
\end{align}
A: Recall that the product $A\vec c$ can be interpreted as the linear combination of the colums $\vec x_i$ of $A$ by the coordinates $c_i$ of $\vec c$
$$A\vec c =\sum c_i\vec x_i$$
Refer also to the related


*

*Matrix multiplication - Express a column as a linear combination
