Linear transformaton and inverse

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Homework Help Overview

The discussion revolves around the properties of linear transformations, specifically focusing on the conditions under which the inverse of an injective linear transformation is also a linear transformation. The subject area is linear algebra, particularly the concepts of injectivity, surjectivity, and bijectivity.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of injectivity for the existence of an inverse linear transformation, with some asserting that injectivity alone is insufficient and that bijectivity is required. Others question the necessity of surjectivity and discuss the relationship between injectivity and surjectivity in establishing bijectivity.

Discussion Status

The discussion is active, with participants providing differing viewpoints on the requirements for a linear transformation's inverse to be linear. Some guidance has been offered regarding the need for surjectivity, and there is an ongoing exploration of the definitions and properties involved.

Contextual Notes

There is a mention of T(V) being a subspace of W, which may influence the discussion on the properties of the transformation. Participants are also addressing the completeness of the original statement regarding the transformation's properties.

autre
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Homework Statement



If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.

The Attempt at a Solution



Let w1, w2 in W. If w1=T(v1) and w2=T(v2), v1=/=v2 in V. Thus, T^-1: V →W is a function. Then, v1+v2=T^-1(w1) + T^-1(w2) and for a in F, T^-1(w1) = aT^-1(w1) = av1 for w in W.
 
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autre said:
If T : V → W is an injective linear transformation, then T^-1: V →W is a linear transformation.

This is not true. Injectivity is not enough. You need bijectivity.
 
Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.
 
micromass said:
This is not true. Injectivity is not enough. You need bijectivity.

but doesn't he have to use injectivity and surjectivity to show bijectivity?
 
autre said:
Oh sorry, I left out a part. T:V -> W is a linear transformation, T(V) is a subspace of W.

Not enough. You need T to be surjective.

mtayab1994 said:
but doesn't he have to use injectivity and surjectivity to show bijectivity?

Yes, but he didn't state that T is surjective.
 
what IS true, is that if T:V→W is an injective linear transformation, then

T-1:T(V)→V is a linear transformation.

now, prove this by explicitly defining what T-1 has to be.
 

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