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Definition: Invertibility of Linear Transformation

Let $V $ and $W $ be vector spaces, and let $T: V \to W $ be linear. A function $U: W \to V$ is said to be an inverse of $T $ if $TU = I_W $ and $UT = I_V $. If $T$ has an inverse, then $T $ is said to be invertible.

If $T$ is invertible, then the inverse of $T $ is unique and is denoted by $\inv{T} $.

The following facts hold for invertible functions $T $ and $U $.

  1. $\inv{(TU)} = \inv{ U } \inv{ T } $
  2. $\inv{(\inv{ T })} = T $
    In particular, $\inv{ T } $ is invertible.
  3. $V $ and $W $ are two finite dimensional spaces of equal dimension. $T: V \to W$ is linear.
    $T $ is invertible $\iff \rank{ T } = \dim V$
Theorem 2.17

Let $V $ and $W $ be vector spaces, and let $T: V \to W $ be linear and invertible. Then $\inv{ T }: W \to V $ is linear.

Definition: Invertiblility of Matrices

Let $A $ an $n \times n $ matrix. Then $A $ is invertible if there exists an $n \times n $ matrix $B $ such that $AB = BA = I $.

If $A $ is invertible, then the matrix $B $ such that $AB = BA = I $ is unique. (If $C $ were another such matrix, then $C = CI = C(AB) = (CA)B = IB = B $.) The matrix $B $ is called the inverse of $A $ and is denoted by $\inv{ A } $.

Lemma

Let $T $ be an invertible linear transformation from $V $ to $W $. Then $V $ is finite-dimensional if and only if $W $ is finite-dimensional. In this case, $\dim V = \dim W $.

Theorem 2.18

Let $V $ and $W $ be finite-dimensional vector spaces with ordered bases $\beta $ and $\gamma $, respectively. Let $T: V \to W $ be linear. Then $T $ is invertible if and only if $[ T ]_{ \beta }^{ \gamma } $ is invertible. Furthermore, $[ \inv{ T } ]_{ \gamma }^{ \beta } = \inv{([ T ]_{ \beta }^{ \gamma })} $.

Corollary 1

Let $V $ be a finite-dimensional vector space with an ordered basis $\beta$, and let $T: V \to V $ be linear. Then $T $ is invertible if and only if $[ T ]_{ \beta } $ is invertible. Furthermore, $[ \inv{ T } ]_{ \beta } = \inv{([ T ]_{ \beta })} $.

Corollary 2

Let $A $ be an $n \times n $ matrix. Then $A $ is invertible if and only if $L_A $ is invertible. Furthermore, $\inv{(L_A)} = L_{ \inv{ A }}$

Definition: Isomorphism

Let $V $ and $W $ be vector spaces. We say that $V$ is isomorphic to $W $ if there exists a linear transformation $T: V \to W $ that is invertible. Such a linear transformation is called an isomorphism from $V $ onto $W $.

Theorem 2.19

Let $V $ and $W $ be finite-dimensional vector spaces (over the same field). Then $V$ is isomorphic to $W $ if and only if $\dim V = \dim W $.

Corollary

Let $V $ be a vector space over $F $. Then $V $ is isomorphic to $F^n $ if and only if $\dim V = n$.

Theorem 2.20

Let $V $ and $W $ be finite-dimensional vector spaces over $F $ of dimensions $n $ and $m $, respectively, and let $\beta $ and $\gamma $ be ordered bases for $V $ and $W $, respectively, then

the function $\Phi: \L(V, W) \to M_{m \times n}(F)$, defined by $\Phi(T) = [ T ]_{ \beta }^{ \gamma } $ for $T \in \L(V, W) $, is an isomorphism.

Corollary

Let $V $ and $W $ be finite-dimensional vector spaces of dimensions $n $ and $m $, respectively. Then $\L(V, W)$ is finite-dimensional of dimension $mn$.

Definition: Standard Representation of Vector Spaces

Let $\beta$ be an ordered basis for an $n $-dimensional vector space V over the field $F $. The standard representation of $V $ with respect to $\beta $ is the function $\phi_ \beta: V \to F^n $ defined by $\phi_ \beta(x) = [x]_ \beta $ for each $x \in V$.

Theorem 2.21

For any finite-dimensional vector space $V $ with ordered basis $\beta , \phi_ \beta $ is an isomorphism.