Does this result for inverses of linear transformation holds for infinite dimensional case.
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Q.If V is an arbitrary vector space over $F$ and if $T$ belongs $Hom(V,V)$ is right-invertible with a unique right inverse, prove that its invertible. This was question asked in Topics in Algebra by I.N. Herstein.
I was able to prove this result by using the argument by contradiction, let us assume $T$ is not invertible and So there exist a $Snot=0$ such that $ST=TS=0$ and let $S_1$ is unique right inverse, now $ S_1TS=IS=S=0$, which is a contradiction. I had question if the vector space is not finite dimensional does the statement holds true? If yes, does my argument above holds.
linear-algebra linear-transformations
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Q.If V is an arbitrary vector space over $F$ and if $T$ belongs $Hom(V,V)$ is right-invertible with a unique right inverse, prove that its invertible. This was question asked in Topics in Algebra by I.N. Herstein.
I was able to prove this result by using the argument by contradiction, let us assume $T$ is not invertible and So there exist a $Snot=0$ such that $ST=TS=0$ and let $S_1$ is unique right inverse, now $ S_1TS=IS=S=0$, which is a contradiction. I had question if the vector space is not finite dimensional does the statement holds true? If yes, does my argument above holds.
linear-algebra linear-transformations
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Q.If V is an arbitrary vector space over $F$ and if $T$ belongs $Hom(V,V)$ is right-invertible with a unique right inverse, prove that its invertible. This was question asked in Topics in Algebra by I.N. Herstein.
I was able to prove this result by using the argument by contradiction, let us assume $T$ is not invertible and So there exist a $Snot=0$ such that $ST=TS=0$ and let $S_1$ is unique right inverse, now $ S_1TS=IS=S=0$, which is a contradiction. I had question if the vector space is not finite dimensional does the statement holds true? If yes, does my argument above holds.
linear-algebra linear-transformations
Q.If V is an arbitrary vector space over $F$ and if $T$ belongs $Hom(V,V)$ is right-invertible with a unique right inverse, prove that its invertible. This was question asked in Topics in Algebra by I.N. Herstein.
I was able to prove this result by using the argument by contradiction, let us assume $T$ is not invertible and So there exist a $Snot=0$ such that $ST=TS=0$ and let $S_1$ is unique right inverse, now $ S_1TS=IS=S=0$, which is a contradiction. I had question if the vector space is not finite dimensional does the statement holds true? If yes, does my argument above holds.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Nov 14 at 15:37
Ash Pd
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In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
add a comment |
up vote
1
down vote
In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
add a comment |
up vote
1
down vote
up vote
1
down vote
In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)
In general the statement is not true. The problem is that $S_1T$ is the identity on the image of $T$ which could be smaller than $V$. Think for example in the space of polynomial ring the multiplication by $x$ which is linear and has an inverse in the image (i.e. divide by $x$ by shifting all the coefficient) but clearly is not invertible on the all space (a polynomial of degree zero has not pre image)
answered Nov 14 at 15:53
ALG
55413
55413
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
add a comment |
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
This provide some light, but could uniqueness of right inverse have some effect on the statement in infinite dimensional case?
– Ash Pd
Nov 15 at 11:42
add a comment |
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