21. Relationship between different bases

We survey all bases of the vector space V. According to Theorem 8, a system of n vectors f₁,···,f_n of V is a base if only if the vectors are linearly independent. The f₁ can be represented in the form

in terms of the base vectors e₁, ··· , e_n, employing certain numbers a_ik as coefficients.

Since the f₁,···,f_n are linearly dependent, there exist n, not all non-zero numbers l₁,···,l_n with the property

l₁f₁ + ··· +l_nf_n = 0.

Substituting here from (21.1), you find

Since the e₁, ··· , e_n are linearly independent, this relationship can only hold if all the coefficients vanish, that is,

These equations can be interpreted as a system of n linear equations for l₁,···,l_n, the right hand sides of which vanish. If these equations hold without all l_i vanishing, then, according to the observation at the end of Section 12, the determinant |A| of the matrix A = (a_ik) in (21.1) must have the value zero.

Conversely, if f₁,···,f_n are linearly independent, they form a base for V. In particular, e₁, ··· , e_n can then be expressed in terms of f₁,···,f_n :

If you substitute in these equations from (21.1), you find

Since the e₁, ··· , e_n are linearly independent, you obtain

This equation leads you to the matrix equation BA = E with B=(b_lm). It says: B is the inverse of A, that is, B = A^-1. In particular, because A^-1 exists, the determinant |A| 0, whence follows

Theorem 9

Vectors f₁,···,f_n form a base of V if and only if the determinant |A| of the matrix (a_ik) in (21.1) is non-zero.

The rows of any (n,p)-matrix M can be interpreted as vectors or a vector space consisting of (1,p) matrices, whence one can speak of linear dependence or independence of the rows of M. A corresponding result applies to its columns.

You draw from (21.1) and (21.2) the conclusion that the rows of the matrix A are then and only then linearly dependent when this is the case for the vectors f₁,···,f_n . More over, Theorem 9 yields: |A| is zero when the rows of A are linearly dependent. If you take still Theorem 4 into consideration, you obtain

Theorem 10

|A| = 0 if and only if the rows (columns) of A are linearly dependent.

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