Linear Algebra/Linear Dependance 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]

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]

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]

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.