Unformatted text preview: B is a transitive GL ( V )set. Solution. Absent. 4.41 (i) Let U = { ( a , a ) : a ∈ R } . Find all the complements of U in R 2 . Solution. If v = ( b , c ) is any nonzero vector with b ±= c , then W v = { a v : a ∈ R } is a complement of U . Moreover, every complement is of this form. (ii) If U is a subspace of a f nite dimensional vector space V , prove that any two complements of U are isomorphic. Solution. Absent. 4.42 If A is an m × n matrix and B is an p × m matrix, prove that rank ( B A ) ≤ rank ( A ). Solution. Absent. 4.43 Let R n be equipped with the usual inner product: if v = ( a 1 , . . . , a n ) and u = ( b 1 , . . . , b n ) , then (v, u ) = a 1 b 1 + ··· + a n b n . (i) Prove that every orthogonal transformation is nonsingular. Solution. Absent....
View
Full Document
 Fall '11
 KeithCornell
 Linear Algebra, Vector Space, Linear map, Isomorphism

Click to edit the document details