Abstract Nonsense

Crushing one theorem at a time

Invertible Linear Transformations (Pt. I)


Point of post: In this post we talk about the equivalent of Halmos’s section 36, but in a more general setting.

Motivation

Recall that in our last post we discussed how to turn \text{End}\left(\mathscr{V}\right) into an associative unital algebra by defining a ‘multiplication map’

\mu:\text{End}\left(\mathscr{V}\right)\times\text{End}\left(\mathscr{V}\right)\to\text{End}\left(\mathscr{V}\right):\left(T,T'\right)\mapsto T\circ T'

We saw though that this multiplication has some downsides (it had zero divisors and wasn’t, in general, commutative).  There is a nice property of this algebra which isn’t enjoyed by all algebras. In particular, it is not ‘uncommon’ for elements of \text{End}\left(\mathscr{V}\right) to have multiplicative inverses, in the usual sense.  Now, at first glance this doesn’t seem that ‘great’ of a property; I mean, not all of  the elements of \text{End}\left(\mathscr{V}\right) have multiplicative inverses just some do. One may expect that, in general, this is a common occurrence among algebras, even more common among associative unital algebras. In fact, this isn’t the case. To see how badly the existence of multiplicative inverses can get screwed up consider for example the polynomial ring \mathbb{R}[x]. It’s fairly plain to see that \mathbb{R}[x] is, in fact, an associative unital algebra over \mathbb{R} with the usual polynomial addition and multiplication. That said, a little thought shows that p(x)\in\mathbb{R}[x] has a multiplicative inverse only if p(x)\in\text{span}\{1\}.

Thus, this post will explore this ‘nice’ quality of the multiplication map on \text{End}\left(\mathscr{V}\right)

Invertible Elements In Associative Unital Algebras

Let \mathscr{A} be an associative unital algebra with identity element \mathbf{1}. We say that x\in\mathscr{A} is invertible if there exists some y\in\mathscr{A} for which

xy=yx=\mathbf{1}

in which case we call y a (soon to be the) inverse of x.  We denote the set of all invertible elements of \mathscr{A} by \mathscr{A}^{\times}. Some theorems come immediately from this definition, first and foremost:

Theorem: Let \mathscr{A} be an associative unital algebra with identity \mathbf{1} and let x\in\mathscr{A}^{\times}. Then, if y and z are both inverses of x then y=z.

Proof: We merely note that by definition xz=\mathbf{1} and so y(xz)=y and by associativity (yx)z=y but by assumption that yx=\mathbf{1} this implies that z=\mathbf{1}z=y from where the conclusion follows. \blacksquare

Remark: Now that we know that for x\in\mathscr{A}^{\times} the inverse of x is unique we may unambiguously denote it by x^{-1}.

The next logical thing to ask is does x,y\in\mathscr{A}^{\times} imply that xy is invertible? What about \alpha x for \alpha\in F? And x^{-1}? We take care of these three things in the next theorem:

Theorem: Let \mathscr{A} be an associative unital F-algebra with identity \mathbf{1}. Then, if x,y\in\mathscr{A}^{\times} then \alpha x,x^{-1},xy\in\mathscr{A} where \alpha\in F-\{0\}.

Proof: To prove that \alpha x\in\mathscr{A}^{\times} it suffices to find an inverse for it. To do this we note that since \alpha\in F-\{0\} that \alpha has an inverse (in the sense of the field operations of F) given by \alpha^{-1}. We merely note then that

(\alpha x)(\alpha^{-1}x^{-1})=(\alpha\alpha^{-1})(xx^{-1})=1\mathbf{1}=\mathbf{1}

To prove that x^{-1}\in\mathscr{A}^{\times} we notice that by definition

xx^{-1}=x^{-1}x=\mathbf{1}

and so x^{-1}\in\mathscr{A}^{\times} and \left(x^{-1}\right)^{-1}=x. Lastly, to prove that xy\in\mathscr{A}^{\times} we note that

(xy)\left(y^{-1}x^{-1}\right)=x\left(yy^{-1}\right)x^{-1}=x\mathbf{1}x^{-1}=xx^{-1}=\mathbf{1}

from where the conclusion follows. \blacksquare

Of course, one may wonder whether x,y\in\mathscr{A}^{\times} implies that x+y\in\mathscr{A}^{\times} so that \mathscr{A}^{\times} becomes a contender to be a linear subspace (and in fact, a subalgebra) of \mathscr{A}. The answer is unfortunately no. Note that \mathbf{0} is not invertible since \mathbf{0}\mathbf{x}=\mathbf{0}\ne\mathbf{1} for all x\in\mathscr{A}. But, if x\in\mathscr{A}^{\times} the above implies that -x\in\mathscr{A}^{\times}, yet by what was just said we know that x+-x=\mathbf{0}\notin\mathscr{A}^{\times}.

Invertible Linear Homomorphisms

Recalling that if \mathscr{V} is an n-dimensional F-space then \text{End}\left(\mathscr{V}\right) is an associative unital algebra with multiplication given by function composition and identity element \text{id}_{\mathscr{V}} we may apply the discussion in the previous to \text{End}\left(\mathscr{V}\right). Recalling though the definition of the multiplication and identity in \text{End}\left(\mathscr{V}\right) we see that T\in\text{End}\left(\mathscr{V}\right) is invertible if and only if there exists some S\in\text{End}\left(\mathscr{V}\right) such that

T\circ S=S\circ T=\text{id}_{\mathscr{V}}

Thus, T is invertible if and only if T has a set-theoretic inverse which is also a linear transformation. Recall though that we proved in an earlier post that if a linear transformation possesses a set-theoretic inverse, denoted T^{-1}, it is a linear transformation. Thus, we are left with the following satisfying theorem:

Theorem: Let \mathscr{V} be an n-dimensional F-space, then T\in\text{End}\left(\mathscr{V}\right) is invertible if and only if T possesses a set-theoretic inverse. Moreover, if T is invertible it’s inverse is the set-theoretic inverse T^{-1}.

In light of the above theorem it’s evident that the study of when a linear transformation T possesses a set-theoretic inverse is  crucial to the study of when T is invertible. So, before we start this study we introduce some terminology. If T\in\text{End}\left(\mathscr{V}\right) is injective we say that T is a monomorphism. If T is surjective we call it an epimorphism. Finally, if T is bijective we call T an isomorphism. Since the concept comes up a lot we denote the set of all isomorphisms on \mathscr{V} by \text{GL}\left(\mathscr{V}\right), and thus to point out the obvious; with this definition \left[\text{End}\left(\mathscr{V}\right)\right]^{\times}=\text{GL}\left(\mathscr{V}\right). Lastly, for T\in\text{End}\left(\mathscr{V}\right) we denote \ker T=\left\{v\in T:T(v)=\bold{0}\right\}.

Remark: It is also common to denote what we called \text{GL}\left(\mathscr{V}\right) by \text{Aut}\left(\mathscr{V}\right). But, this leaves confusion when dealing with \text{Aut}\left(G\right) for a group G.

We note from our initial section that \text{GL}\left(\mathscr{V}\right) is closed under inversion, multiplication, and scalar multiplication. In fact, realizing that \mathbf{1}\in\text{GL}\left(\mathscr{V}\right) pretty much proves that \text{GL}\left(\mathscr{V}\right) is a group under multiplication. So, we begin our characterization of these maps

Theorem: Let T\in\text{End}\left(\mathscr{V}\right). Then, T is a monomorphism if and only if \ker T=\{\bold{0}\}.

Proof: First suppose that T is a monomorphism then since T(\bold{0})=\bold{0} we know that \{\bold{0}\}=T^{-1}(\{\bold{0}\})=\ker T.

Conversely, suppose that \ker T=\{\bold{0}\}. Then we see that T(x)=T(y) implies T(x)-T(y)=\bold{0} and since T is a linear transformation we may conclude that T(x-y)=\bold{0} and thus x-y=\bold{0}, or x=y.

The conclusion follows. \blacksquare

Remark: Note that we did not use the ‘full extent’ of the linearity of T in the sense that we didn’t use the fact that T(\alpha x)=\alpha T(x). This is because the above theorem holds for homomorphisms between groups and every vector space is an abelian group.

References:

1. Halmos, Paul R.  Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print

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November 30, 2010 - Posted by | Algebra, Halmos, Linear Algebra | , , , , , , ,

7 Comments »

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