Bilinear Forms Definition 10. A bilinear form B : (V, W ) ? R is said to be nond
ID: 1947889 • Letter: B
Question
Bilinear FormsDefinition 10. A bilinear form B : (V, W ) ? R is said to be nondegenerate provided that
1. B(v,w)=0forallw?W iffv=0,and 2. B(v,w)=0forallv?V iffw=0.
Problem 4.2. Prove that the bilinear form on Rn given by the standard inner product is
nondegenerate.
Explanation / Answer
Every bilinear form B on V defines a pair of linear maps from V to its dual space V*. Define by B1(v)(w)=B(v,w) B(v)(w)=B(v,w) This is often denoted as B1(v)=B(v,.) B2(v)=B(.,v) where the (.) indicates the slot into which the argument for the resulting linear functional is to be placed. If either of B1 or B2 is an isomorphism, then both are, and the bilinear form B is said to be nondegenerate. This can only occur if V is finite-dimensional since V* has higher dimension than V otherwise. If V is finite-dimensional then one can identify V with its double dual V**. One can then show that B2 is the transpose of the linear map B1 (if V is infinite-dimensional then B2 is the transpose of B1 restricted to the image of V in V**). Given B one can define the transpose of B to be the bilinear form given by B*(v,w)=B(w,v) If V is finite-dimensional then the rank of B1 is equal to the rank of B2. If this number is equal to the dimension of V then B1 and B2 are linear isomorphisms from V to V*. In this case B is nondegenerate. By the rank-nullity theorem, this is equivalent to the condition that the kernel of B1 be trivial. In fact, for finite dimensional spaces, this is often taken as the definition of nondegeneracy. Thus B is nondegenerate if and only if B(v,w)=0 for all w=>v=0 Given any linear map A : V ? V* one can obtain a bilinear form B on V via B(v,w)=A(v)(w) This form will be nondegenerate if and only if A is an isomorphism. If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the associated matrix is non-singular.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.