Invertible Linear Transformations

In this section we will conclude our introduction to linear transformations by bringing together the twin properties of injectivity and surjectivity and consider linear transformations with both of these properties.

1. easily
2. What does it mean to say two vector spaces are isomorphic? Both technically, and informally?
3. How do linear transformations relate to systems of linear equations?

One preliminary definition, and then we will have our main definition for this section.

Informally, $I_W$ is the do-nothing function. You should check that $I_W$ is really a linear transformation, as claimed, and then compute its kernel and range to see that it is both injective and surjective. All of these facts should be straightforward to verify (). With this in hand we can make our main definition.

Informally, a linear transformation $T$ is invertible if there is a companion linear transformation, $S$, which undoes the action of $T$. When the two linear transformations are applied consecutively (composition), in either order, the result is to have no real effect. It is entirely analogous to squaring a positive number and then taking its (positive) square root.

Here is an example of a linear transformation that is invertible. As usual at the beginning of a section, do not be concerned with where $S$ came from, just understand how it illustrates .

It can be as instructive to study a linear transformation that is not invertible.

In you may have noticed that $T$ is not surjective, since the matrix $A$ was not in the range of $T$. And $T$ is not injective since there are two different input column vectors that $T$ sends to the matrix $B$. Linear transformations $T$ that are not surjective lead to putative inverse functions $S$ that are undefined on inputs outside of the range of $T$. Linear transformations $T$ that are not injective lead to putative inverse functions $S$ that are multiply-defined on each of their inputs. We will formalize these ideas in .

But first notice in that we only require the inverse (when it exists) to be a function. When it does exist, it too is a linear transformation.

So when $T$ has an inverse, $\ltinverse{T}$ is also a linear transformation. Furthermore, $\ltinverse{T}$ is an invertible linear transformation and its inverse is what you might expect.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

We now know what an inverse linear transformation is, but just which linear transformations have inverses? Here is a theorem we have been preparing for all chapter long.

Many will call an injective and surjective function a bijective function or just a bijection. tells us that this is just a synonym for the term invertible (which we will use exclusively).

We can follow the constructive approach of the proof of to construct the inverse of a specific linear transformation, as the next example shows.

We will make frequent use of the characterization of invertible linear transformations provided by . The next theorem is a good example of this, and we will use it often, too.

When a composition is invertible, the inverse is easy to construct.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

A vector space is defined () as a set of objects (vectors) endowed with a definition of vector addition ($+$) and a definition of scalar multiplication (written with juxtaposition). Many of our definitions about vector spaces involve linear combinations (), such as the span of a set () and linear independence (). Other definitions are built up from these ideas, such as bases () and dimension (). The defining properties of a linear transformation require that a function respect the operations of the two vector spaces that are the domain and the codomain (). Finally, an invertible linear transformation is one that can be undone it has a companion that reverses its effect. In this subsection we are going to begin to roll all these ideas into one.

A vector space has structure derived from definitions of the two operations and the requirement that these operations interact in ways that satisfy the ten properties of . When two different vector spaces have an invertible linear transformation defined between them, then we can translate questions about linear combinations (spans, linear independence, bases, dimension) from the first vector space to the second. The answers obtained in the second vector space can then be translated back, via the inverse linear transformation, and interpreted in the setting of the first vector space. We say that these invertible linear transformations preserve structure. And we say that the two vector spaces are structurally the same. The precise term is isomorphic, from Greek meaning of the same form. Let us begin to try to understand this important concept.

A few comments on this definition. First, be careful with your language (). Two vector spaces are isomorphic, or not. It is a yes/no situation and the term only applies to a pair of vector spaces. Any invertible linear transformation can be called an isomorphism, it is a term that applies to functions. Second, given a pair of vector spaces there might be several different isomorphisms between the two vector spaces. But it only takes the existence of one to call the pair isomorphic. Third, $U$ isomorphic to $V$, or $V$ isomorphic to $U$? It does not matter, since the inverse linear transformation will provide the needed isomorphism in the opposite direction. Being isomorphic to is an equivalence relation on the set of all vector spaces (see for a reminder about equivalence relations).

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Just as a matrix has a rank and a nullity, so too do linear transformations. And just like the rank and nullity of a matrix are related (they sum to the number of columns, ) the rank and nullity of a linear transformation are related. Here are the definitions and theorems, see the Archetypes () for loads of examples.

Here are two quick theorems.

Just as injectivity and surjectivity come together in invertible linear transformations, there is a clear relationship between rank and nullity of a linear transformation. If one is big, the other is small.

said that the rank and nullity of a matrix sum to the number of columns of the matrix. This result is now an easy consequence of when we consider the linear transformation $\ltdefn{T}{\complex{n}}{\complex{m}}$ defined with the $m\times n$ matrix $A$ by $\lt{T}{\vect{x}}=A\vect{x}$. The range and kernel of $T$ are identical to the column space and null space of the matrix $A$ (, ), so the rank and nullity of the matrix $A$ are identical to the rank and nullity of the linear transformation $T$. The dimension of the domain of $T$ is the dimension of $\complex{n}$, exactly the number of columns for the matrix $A$.

We should mention that the notation T^-1 will yield an inverse of a linear transformation in Sage.

Also, the rank and nullity are what you might expect. Recall that for a matrix Sage provides a left nullity and a right nullity. There is no such distinction for linear transformations. We verify as an example.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

This subsection does not really belong in this section, or any other section, for that matter. It is just the right time to have a discussion about the connections between the central topic of linear algebra, linear transformations, and our motivating topic from , systems of linear equations. We will discuss several theorems we have seen already, but we will also make some forward-looking statements that will be justified in .

and are ideal examples to illustrate connections with linear transformations. Both have the same coefficient matrix, D=

To apply the theory of linear transformations to these two archetypes, employ the matrix-vector product () and define the linear transformation, \ltdefn{T}{\complex{4}}{\complex{3}},\quad \lt{T}{\vect{x}}=D\vect{x} =x_1\colvector{2\\-3\\1}+ x_2\colvector{1\\4\\1}+ x_3\colvector{7\\-5\\4}+ x_4\colvector{-7\\-6\\-5}

tells us that $T$ is indeed a linear transformation. asks for solutions to $\linearsystem{D}{\vect{b}}$, where $\vect{b}=\colvector{8\\-12\\-4}$. In the language of linear transformations this is equivalent to asking for $\preimage{T}{\vect{b}}$. In the language of vectors and matrices it asks for a linear combination of the four columns of $D$ that will equal $\vect{b}$. One solution listed is $\vect{w}=\colvector{7\\8\\1\\3}$. With a nonempty preimage, tells us that the complete solution set of the linear system is the preimage of $\vect{b}$, \vect{w}+\krn{T}=\setparts{\vect{w}+\vect{z}}{\vect{z}\in\krn{T}}

The kernel of the linear transformation $T$ is exactly the null space of the matrix $D$ (see ), so this approach to the solution set should be reminiscent of . The kernel of the linear transformation is the preimage of the zero vector, exactly equal to the solution set of the homogeneous system $\homosystem{D}$. Since $D$ has a null space of dimension two, every preimage (and in particular the preimage of $\vect{b}$) is as big as a subspace of dimension two (but is not a subspace).

is identical to but with a different vector of constants, $\vect{d}=\colvector{2\\3\\2}$. We can use the same linear transformation $T$ to discuss this system of equations since the coefficient matrix is identical. Now the set of solutions to $\linearsystem{D}{\vect{d}}$ is the pre-image of $\vect{d}$, $\preimage{T}{\vect{d}}$. However, the vector $\vect{d}$ is not in the range of the linear transformation (nor is it in the column space of the matrix, since these two sets are equal by ). So the empty pre-image is equivalent to the inconsistency of the linear system.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.