Linear Algebra/Linear Dependance of Columns

Let C₁, C₂, C₃, ..., C_n be n columns of m numbers $C_n = \begin{bmatrix} a_{1n} \\ a_{2n} \\ a_{3n} \\ \vdots \\ a_{mn} \\ \end{bmatrix}$.

A linear combination of columns n₁C₁+n₂C₂+n₃C₃+...+n_nC_n is the column

$C_n = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ \vdots \\ c_n \\ \end{bmatrix}$.

Where c_k=n₁a_k1+n₁a_k1+n₂a_k2+n₃a_k3+...+n_na_kn.

Theorem

If there is a determinant of order n which is A=a_ij, and there are n columns of n elements such that the ith entry of the jth column is equal to a_ij, then if one of the columns is a linear combination of the other columns, then the determinant is equal to 0.

Proof

Suppose that the kth column is a linear combination of the other column,

$\begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & c_1a_{11}+c_2a_{12}+ c_3a_{13} + \ldots + c_na_1n & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & c_1a_{21}+c_2a_{22}+ c_3a_{23} + \ldots + c_na_2n & \ldots & a_{2n} \\ a_{31} & a_{23} & a_{33} & \ldots & c_1a_{31}+c_2a_{32}+ c_3a_{33} + \ldots + c_na_3n & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n3} & a_{n3} & \ldots & c_1a_{n1}+c_2a_{n2}+ c_3a_{n3} + \ldots + c_na_nn & \ldots & a_{nn} \\ \end{bmatrix}$.

Then by the linearity of determinants, the determinant is equal to

$c_1\begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{11} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{21} & \ldots & a_{2n} \\ a_{31} & a_{23} & a_{33} & \ldots & a_{31} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n3} & a_{n3} & \ldots & a_{n1} & \ldots & a_{nn} \\ \end{bmatrix} + c_2\begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{22} & \ldots & a_{2n} \\ a_{31} & a_{23} & a_{33} & \ldots & a_{32} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n3} & a_{n3} & \ldots & a_{n2} & \ldots & a_{nn} \\ \end{bmatrix} + c_3\begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{13} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{23} & a_{33} & \ldots & a_{33} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n3} & a_{n3} & \ldots & a_{n3} & \ldots & a_{nn} \\ \end{bmatrix} + \ldots + c_n\begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} & \ldots & a_{2n} \\ a_{31} & a_{23} & a_{33} & \ldots & a_{3n} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{n1} & a_{n3} & a_{n3} & \ldots & a_{nn} & \ldots & a_{nn} \\ \end{bmatrix}$.

Since all of those matrices have repeat columns, their determinants are 0, and so their sum is 0.

Rank of a Matrix

The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the order of the rank of the matrix is called a basis minor of the matrix, and the columns that the minor includes are called the basis columns.