LINEAR ALGEBRA
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On Z3, Let A = (0 1 0 0 0 1 0 0 1). Let V = {(a b c)|a, b, c epsilon Z, } and f:V ---> V be defined by f(u) = Au. find the kernel of f and show it is a congruence.
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26-Feb-2015 at 6:00 am.
-------------------------On Z3, Let A = (0 1 0 0 0 1 0 0 1). Let V = {(a b c)|a, b, c epsilon Z, } and f:V ---> V be defined by f(u) = Au. find the kernel of f and show it is a congruence.
| A= | V= | ||||||||
| 0 | 1 | 0 | a | ||||||
| 0 | 0 | 1 | b | ||||||
| 0 | 0 | 1 | c | ||||||
| F[U] = A*U | |||||||||
| KERNEL IS GIVEN BY | |||||||||
| F[U] = A*U =0 | |||||||||
| 0 | 1 | 0 | a | 0 | |||||
| 0 | 0 | 1 | * | b | = | 0 | |||
| 0 | 0 | 1 | c | 0 | |||||
| b=0….c=0…. | |||||||||
| a CAN BE ANY VALUE …SO KERNEL IS … | |||||||||
| 1 | |||||||||
| 0 | |||||||||
| 0 | …………..ANSWER | ||||||||
| .. | |||||||||
| F[U]=A*U ………………. | |||||||||
| HERE THE KERNEL OF THIS TRANSFORM MEANS THE | |||||||||
| EQUIVALENCE RELATION THAT F INDUCES ON ITS DOMAIN …THAT IS … | |||||||||
| IF U = U ' ….THEN IT IMPLIES THAT F[U] = F[U'] | |||||||||
| HENCE WE SAY THE KERNEL OF THE L.T. | |||||||||
| IS A CONGRUENCE RELATION | |||||||||
posted by mathsmanagementcommonsense at 8:40 PM
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