2. Which of the following are bilinear forms on
V2 (R)?
Let x = (x1, x2) and y = (y1, y2).
Answers
Answer:
Fp�Fq� � �Fs is precisely of the form M(x, y, ... , t) = S1� i� p S1� j� q � S1� l� s aij�l xiyj � tl.
3. Let V1 , V2 , � , Vk be finite dimensional vector spaces over a field F, with ordered bases b 1, b 2, � , b k and the respective dual bases f 1, f 2, � , f k. Using these describe a multilinear expression M(v1, v2, � , vk) on V1�V2 � � � Vk , explicitly in terms of the vectors v1, v2, � , vk.
4. If V and W are finite dimensional vector spaces and B(v, w) is a non-zero bilinear expression on V�W, show that there exists a positive integer k and ordered bases b and g of V and W such that if x = [v]b and y = [w]g then B(v, w) = S1� i� k xiyi, v Î V, w Î W. This result could be restated as
[B]b g = .
5. If V and W are finite dimensional vector spaces over a field F, the rank of a bilinear expression B(v, w) on V�W is defined to be the rank of a representing
Step-by-step explanation:
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