three_ii.pdf

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

• Homework Help
• someoneonearth
• 71

This preview shows pages 1–11. Sign up to view the full content.

Three.II Homomorphisms Linear Algebra Jim Hefferon

This preview has intentionally blurred sections. Sign up to view the full version.

Definition
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

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
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 )

This preview has intentionally blurred sections. Sign up to view the full version.

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
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).

This preview has intentionally blurred sections. Sign up to view the full version.

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 ) .
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 )

This preview has intentionally blurred sections. Sign up to view the full version.

Example The derivative is the usual transformation of polynomial space d/dx : P 2 P 1 .
This is the end of the preview. Sign up to access the rest of the document.
• Fall '16
• Darij Grinberg

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern