# Singular

#### D.3.1.30 exteriorBasis

Procedure from library `matrix.lib` (see matrix_lib).

Return:
qring, an exterior algebra containing the ideal "extBasis", being a basis of the k-th exterior power of an n-dim vector space.

Note:
The output polynomial ring has characteristics 0 and n variables named "S(i)", where the base variable name S is either given by the optional string argument(which must not contain brackets) or equal to "e" by default.

Example:
 ```LIB "matrix.lib"; // basis of the 3-rd symmetricPower of a 4-dim vector space: def r = exteriorBasis(4, 3, "@e"); setring r; r; // container ring: ==> // coefficients: QQ ==> // number of vars : 4 ==> // block 1 : ordering dp ==> // : names @e(1) @e(2) @e(3) @e(4) ==> // block 2 : ordering C ==> // noncommutative relations: ==> // @e(2)@e(1)=-@e(1)*@e(2) ==> // @e(3)@e(1)=-@e(1)*@e(3) ==> // @e(4)@e(1)=-@e(1)*@e(4) ==> // @e(3)@e(2)=-@e(2)*@e(3) ==> // @e(4)@e(2)=-@e(2)*@e(4) ==> // @e(4)@e(3)=-@e(3)*@e(4) ==> // quotient ring from ideal ==> _[1]=@e(4)^2 ==> _[2]=@e(3)^2 ==> _[3]=@e(2)^2 ==> _[4]=@e(1)^2 extBasis; // exterior basis: ==> extBasis[1]=@e(2)*@e(3)*@e(4) ==> extBasis[2]=@e(1)*@e(3)*@e(4) ==> extBasis[3]=@e(1)*@e(2)*@e(4) ==> extBasis[4]=@e(1)*@e(2)*@e(3) ```