Raising the bar

Properties IV-VI

Exercise 1. Let be a linearly independent system of vectors in . Then it can be completed with a vector to form a basis in

Proof. One way to obtain is this. Let be a projector onto the span of  and let Take as any nonzero vector from the image of It is orthogonal to any element of the image of and, in particular, to elements of Therefore completed with gives a linearly independent system . is a basis because for any

Property IV. Additivity. Suppose the th row of is a sum of two vectors:

Denote

,

(except for the th row, all others are the same for all three matrices). Then

Proof. Denote the system of vectors

Case 1. If is linearly dependent, then the system of all rows of is also linearly dependent. By Property III the determinants of all three matrices are zero and the statement is true.

Case 2. Let be linearly independent. Then by Exercise 1 it can be completed with a vector to form a basis in can be represented as linear combinations of elements of We are interested only in the coefficients of  in those representations. So let where and are linear combinations of elements of  Hence,

We can use Property II to eliminate and from the th rows of and respectively, without changing the determinants of those matrices. Let denote the matrix obtained by replacing the th row of with Then by Property II and Axiom 1

which proves the statement.

Combining homogeneity and additivity, we get the following important property that some people use as a definition:

Property V. Multilinearity. The determinant of is a multilinear function of its rows, that is, for each it is linear in row when the other rows are fixed.

Property VI. Antisymmetry. If the matrix is obtained from by changing places of two rows, then

Proof. Let

(all other rows of these matrices are the same). Consider the next sequence of transformations:

By Property II, each of these transformations preserves Recalling homogeneity, we finish the proof.