Linear Algebra/Linear Dependence of Columns

Let C₁, C₂, C₃, ..., C_n be n columns of m numbers $C_n=\left[\begin{array}{c}{a}_{1n}\\ a_{2n}\\ a_{3n}\\ ⋮\\ a_{mn}\end{array}\right]$.

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

$C_n=\left[\begin{array}{c}{c}_1\\ c_2\\ c_3\\ ⋮\\ c_n\end{array}\right]$.

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,

$\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & c_1{a}_{11}+c_2{a}_{12}+c_3{a}_{13}+\dots +c_n{a}_{1}n& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & c_1{a}_{21}+c_2{a}_{22}+c_3{a}_{23}+\dots +c_n{a}_{2}n& \dots & a_{2n}\\ a_{31}& a_{23}& a_{33}& \dots & c_1{a}_{31}+c_2{a}_{32}+c_3{a}_{33}+\dots +c_n{a}_{3}n& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n3}& a_{n3}& \dots & c_1{a}_{n1}+c_2{a}_{n2}+c_3{a}_{n3}+\dots +c_n{a}_{n}n& \dots & a_{nn}\end{array}\right]$.

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

$c_1\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{11}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{21}& \dots & a_{2n}\\ a_{31}& a_{23}& a_{33}& \dots & a_{31}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n3}& a_{n3}& \dots & a_{n1}& \dots & a_{nn}\end{array}\right]+c_2\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{22}& \dots & a_{2n}\\ a_{31}& a_{23}& a_{33}& \dots & a_{32}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n3}& a_{n3}& \dots & a_{n2}& \dots & a_{nn}\end{array}\right]+c_3\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{13}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{23}& \dots & a_{2n}\\ a_{31}& a_{23}& a_{33}& \dots & a_{33}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n3}& a_{n3}& \dots & a_{n3}& \dots & a_{nn}\end{array}\right]+\dots +c_n\left[\begin{array}{ccccccc}{a}_{11}& a_{12}& a_{13}& \dots & a_{1n}& \dots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \dots & a_{2n}& \dots & a_{2n}\\ a_{31}& a_{23}& a_{33}& \dots & a_{3n}& \dots & a_{3n}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ a_{n1}& a_{n3}& a_{n3}& \dots & a_{nn}& \dots & a_{nn}\end{array}\right]$.

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.