# Adv Alegbra HW Solutions 134 - B is a transitive GL V-set...

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134 prove that there exists a unique solution if and only if det ( A ) ±= 0. Solution. Absent. (ii) If V is a vector space with basis X = v 1 ,v 2 ,de f ne T : V V by T (v 1 ) = a v 1 + b v 2 and T (v 2 ) = c v 1 + d v 2 . Prove that T is a nonsingular linear transformation if and only if det ( X [ T ] X ) ±= 0. Solution. Absent. 4.38 Let U be a subspace of a vector space V . (i) Prove that the natural map π : V V / U , given by v 7→ v + U , is a linear transformation with kernel U . Solution. Absent. (ii) State and prove the frst isomorphism theorem for vector spaces. Solution. Here is the statement. If f : V W is a linear trans- formation with ker f = U , then U is a subspace of V and there is an isomorphism ϕ : V / U im f , namely, ϕ(v + U ) = f (v) . 4.39 Let k be a f eld and let k × be its multiplicative group of nonzero ele- ments. Prove that det : GL ( 2 , k ) k × is a surjective group homomor- phism whose kernel is SL ( 2 , k ) . Conclude that SL ( 2 , k ) C GL ( 2 , k ) and GL ( 2 , k )/ SL ( 2 , k ) = k × . Solution. Absent. 4.40 Let V be a f nite dimensional vector space over a f eld k , and let B denote the family of all the bases of V . Prove that
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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....
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