Linear Algebra/Linear Dependence of Columns

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Let C1, C2, C3, ..., Cn be n columns of m numbers .

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


Where ck=n1ak1+n1ak1+n2ak2+n3ak3+...+nnakn.

Theorem[edit | edit source]

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

Proof[edit | edit source]

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


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


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

Rank of a Matrix[edit | edit source]

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.