Linear Algebra/Linear Dependence 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,

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

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.