How can you prove that an operator is an isomorphism? (a) What is another way yo

Question

How can you prove that an operator is an isomorphism? (a) What is another way you can prove that an operator is an isomorphism? (b) Why did I say “operator” for this question, instead of “linear transformation”? (c) Example: Prove that the linear operator T (R2) given by T(x,y) = (y,x) is an isomorphism. Use three different methods. How can you prove that an operator is an isomorphism? (a) What is another way you can prove that an operator is an isomorphism? (b) Why did I say “operator” for this question, instead of “linear transformation”? (c) Example: Prove that the linear operator T (R2) given by T(x,y) = (y,x) is an isomorphism. Use three different methods. How can you prove that an operator is an isomorphism? (a) What is another way you can prove that an operator is an isomorphism? (b) Why did I say “operator” for this question, instead of “linear transformation”? (c) Example: Prove that the linear operator T (R2) given by T(x,y) = (y,x) is an isomorphism. Use three different methods.

Explanation / Answer

Definitions --->

Linear transformation ----Let V,W be two vector spaces over field F .We call a function T :V--------> W alinear transformation from V to W if for all x,y in V and c in F

T(x+c.y)=T(x)+c.T(y)

LINEAR OPERATOR ---- A linear transformation fom V to V is called linear operator

(a) AN linear operator T :V--> V is an isomporphism if T is one one and onto map ( no need to chech that T is linear transformation because given that T is linear operator )

(b) Let T be a linear transformation from V to W , T:V -------> W

if T is an isomorphism ( as in (a) ) then T IS ONTO and hence T(V)=W

And ker(T)={0} taht is nullity T =0 and by rank nullity theorem dim(range T)=dim(V)

and since range T= W then dimW=dimV

HENCE V and W are ismorphic

and this is the reason that we say linear operator since now T:V -------> W becomes T:V --------> V and this is linear operator .

(c) T: R^2-----> R^2 by T(x,y)=(y,x)

to prove T is an isomorphis it is enough to show that kerT is {0}

for , kerT ={( x,y) in R^2: T(x,y)=0}

        ={(x,y) in R^2 : (y,x)=(0,0)}

          = { (x,y) in R^2 :y=0,x=0} =(0,0)

which imples T is an isomorphism .

or in othe way you can check T is one one and onto .

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