three_ii.pdf - Three.II Homomorphisms Linear Algebra Jim...

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Three.II Homomorphisms Linear Algebra Jim Hefferon
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Definition
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Homomorphism 1.1 Definition A function between vector spaces h : V W that preserves addition if ~ v 1 , ~ v 2 V then h ( ~ v 1 + ~ v 2 ) = h ( ~ v 1 ) + h ( ~ v 2 ) and scalar multiplication if ~ v V and r R then h ( r · ~ v ) = r · h ( ~ v ) is a homomorphism or linear map .
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Example Of these two maps h, g : R 2 R , the first is a homomorphism while the second is not. x y h 7-→ 2x - 3y x y g 7-→ 2x - 3y + 1
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Example Of these two maps h, g : R 2 R , the first is a homomorphism while the second is not. x y h 7-→ 2x - 3y x y g 7-→ 2x - 3y + 1 The map h respects addition h ( x 1 y 1 + x 2 y 2 ) = h ( x 1 + x 2 y 1 + y 2 ) = 2 ( x 1 + x 2 ) - 3 ( y 1 + y 2 ) = ( 2x 1 - 3y 1 ) + ( 2x 2 - 3y 2 ) = h ( x 1 y 1 ) + h ( x 2 y 2 ) and scalar multiplication. r · h ( x y ) = r · ( 2x - 3y ) = 2rx - 3ry = ( 2r ) x - ( 3r ) y = h ( r · x y )
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Example Of these two maps h, g : R 2 R , the first is a homomorphism while the second is not. x y h 7-→ 2x - 3y x y g 7-→ 2x - 3y + 1 The map h respects addition h ( x 1 y 1 + x 2 y 2 ) = h ( x 1 + x 2 y 1 + y 2 ) = 2 ( x 1 + x 2 ) - 3 ( y 1 + y 2 ) = ( 2x 1 - 3y 1 ) + ( 2x 2 - 3y 2 ) = h ( x 1 y 1 ) + h ( x 2 y 2 ) and scalar multiplication. r · h ( x y ) = r · ( 2x - 3y ) = 2rx - 3ry = ( 2r ) x - ( 3r ) y = h ( r · x y ) In contrast, g does not respect addition. g ( 1 4 + 5 6 ) = - 17 g ( 1 4 ) + g ( 5 6 ) = - 16
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We proved these two while studying isomorphisms. 1.6 Lemma A homomorphism sends the zero vector to the zero vector. 1.7 Lemma The following are equivalent for any map f : V W between vector spaces. (1) f is a homomorphism (2) f ( c 1 · ~ v 1 + c 2 · ~ v 2 ) = c 1 · f ( ~ v 1 ) + c 2 · f ( ~ v 2 ) for any c 1 , c 2 R and ~ v 1 , ~ v 2 V (3) f ( c 1 · ~ v 1 + · · · + c n · ~ v n ) = c 1 · f ( ~ v 1 ) + · · · + c n · f ( ~ v n ) for any c 1 , . . . , c n R and ~ v 1 , . . . , ~ v n V To verify that a map is a homomorphism the one that we use most often is statement (2).
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We proved these two while studying isomorphisms. 1.6 Lemma A homomorphism sends the zero vector to the zero vector. 1.7 Lemma The following are equivalent for any map f : V W between vector spaces. (1) f is a homomorphism (2) f ( c 1 · ~ v 1 + c 2 · ~ v 2 ) = c 1 · f ( ~ v 1 ) + c 2 · f ( ~ v 2 ) for any c 1 , c 2 R and ~ v 1 , ~ v 2 V (3) f ( c 1 · ~ v 1 + · · · + c n · ~ v n ) = c 1 · f ( ~ v 1 ) + · · · + c n · f ( ~ v n ) for any c 1 , . . . , c n R and ~ v 1 , . . . , ~ v n V To verify that a map is a homomorphism the one that we use most often is statement (2). Example Between any two vector spaces the zero map Z : V W given by Z ( ~ v ) = ~ 0 W is a linear map. Using (2): Z ( c 1 ~ v 1 + c 2 ~ v 2 ) = ~ 0 W = ~ 0 W + ~ 0 W = c 1 Z ( ~ v 1 ) + c 2 Z ( ~ v 2 ) .
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Example The inclusion map ι : R 2 R 3 ι ( x y ) = x y 0 is a homomorphism. ι ( c 1 · x 1 y 1 + c 2 · x 2 y 2 ) = ι ( c 1 x 1 + c 2 x 2 c 1 y 1 + c 2 y 2 ) = c 1 x 1 + c 2 x 2 c 1 y 1 + c 2 y 2 0 = c 1 x 1 c 1 y 1 0 + c 2 x 2 c 2 y 2 0 = c 1 · ι ( x 1 y 1 ) + c 2 · ι ( x 2 y 2 )
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Example The derivative is the usual transformation of polynomial space d/dx : P 2 P 1 .
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