Properties and Examples

Some simple yet important and often used properties of linear transformations are collected in

Theorem 3.

Let \(\ V\ \) and \(\ W\ \) be vector spaces over the field \(\,K.\ \)

If the map \(\ F:\,V\rightarrow W\ \) is a linear transformation, then

1. \(\ F(\theta_V)\ =\ \theta_W\,;\)

2. \(\ F(-\,v)\ =\ -\ F(v)\,,\qquad\forall\ v\in V\,;\)

3. \(\ F(v_1-\,v_2)\ =\ F(v_1)\,-\,F(v_2)\,, \qquad\forall\ \ v_1,\,v_2\in V\,.\)

Proof \(\,\) goes on the relations:

\[\begin{split}\begin{array}{rcl} \theta_V\,=\ 0\cdot\theta_V\,, & \qquad & 0\cdot w\ =\ \theta_W\,, \\ -\ v\ =\ (-1)\cdot v\,, & \qquad & (-1)\cdot w\ =\ -\ w\,, \end{array} \qquad\quad\forall\ v\in V,\ \ \forall\ w\in W.\end{split}\]

Namely, making use of the linearity of the transformation \(\ F\ \) we get:

1. \(\ F(\theta_V)\ =\ F(0\cdot\theta_V)\ =\ 0\cdot F(\theta_V)\ =\ \theta_W\,;\)

2. \(\ F(-\,v)\ =\ F\,[\,(-1)\cdot v\,]\ =\ (-1)\cdot F(v)\ =\ -\ F(v)\,;\)

3. \(\ F(v_1-v_2)\ =\ F\,[\,v_1+(-1)\cdot v_2\,]\ =\ F(v_1)+(-1)\cdot F(v_2)\ =\ F(v_1)-F(v_2)\ \bullet\)

\(\;\)

Being a bijective mapping, an isomorphism is invertible. It is interesting, yet not obvious, that such an inverse of an isomorphism is also a linear transformation, thus an isomorphism.

Theorem 4.

Let \(\ V\ \) and \(\ W\ \) be vector spaces over the field \(\,K.\) \(\\\) If the linear transformation \(\ F:\,V\rightarrow W\ \) is an isomorphism, then the inverse map \(\ F^{-1}:\ W\rightarrow V\ \) is also a linear transformation, hence it is also an isomorphism.

Proof. \(\,\) The map \(\,F^{-1}\,\) being obviously bijective, we only have to validate its additivity and homogeneity.

Taking into account that the map \(\,F\ \) is linear and injective, we get

\[ \begin{align}\begin{aligned}F[F^{-1}(w_1+w_2)]=w_1+w_2=F[F^{-1}(w_1)]+F[F^{-1}(w_2)]= F[F^{-1}(w_1)+F^{-1}(w_2)]\,,\\\text{hence}\qquad F^{-1}(w_1+w_2)\ =\ F^{-1}(w_1)+F^{-1}(w_2)\,;\\\text{}\qquad F[F^{-1}(a\,w)]=a\,w=a\,F[F^{-1}(w)]=F[a\,F^{-1}(w)]\,,\\\text{wherefrom}\qquad F^{-1}(a\,w)= a\,F^{-1}(w)\,,\qquad\forall\ \ w_1,w_2,w\in W,\ \ \forall\ a\in K.\end{aligned}\end{align} \]

Corollary.

The set \(\ \text{Aut}(V)\ \) is a group under composition of mappings.

Indeed, a composition of two automorphisms defined on a space \(\ V(K)\ \) is an automorphism on \(\ V.\ \) The composition itself, as a composition of mappings, is associative. The neutral element is the identity automorphism \(\,I(v)=v,\ \ \forall\ v\in V.\ \) Finally, in virtue of Theorem 4., the inverse of an automorphism is an automorphism. The group \(\ \text{Aut}(V)\ \) is in general non-commutative.

\(\,\)

We still consider the two vector spaces, \(\ V\,\) and \(\,W,\ \) over the field \(\,K.\ \) Assume that \(\ \text{dim}\,V=n\ \) and that the set \(\,B = \{v_1,v_2,\dots,v_n\}\ \) is a basis of the space \(\,V.\ \) Then every vector \(\,v\in V\ \) is represented in a unique way by a linear combination of vectors from \(\,B:\)

\[v\ =\ a_1\,v_1\,+\;a_2\,v_2\,+\ \dots\ +\;a_n\,v_n\,.\]

If \(\,F:\,V\rightarrow W\,\) is a linear transformation, then the image of the vector \(\,v\ \) is given by

\begin{eqnarray*} F(v) & = & F(a_1\,v_1\,+\;a_2\,v_2\,+\ \dots\ +\;a_n\,v_n) \\ & = & a_1\,Fv_1\,+\;a_2\,Fv_2\,+\ \dots\ +\;a_n\,Fv_n\,. \end{eqnarray*}

The last formula reveals an interesting property of the transformation \(\,F,\ \) resulting from the linearity. To determine the image \(\,F(v)\,\) of any from among infinitely many vectors \(\,v\in V,\ \) it is sufficient to know the images of \(\,n\ \) vectors only (provided that these vectors form a basis). This remark is complemented by

Theorem 5. \(\\\)

Let \(\ V\ \) and \(\ W\ \) be vector spaces over the field \(\,K,\ \) while \(\ V\ \) is finite-dimensional one with a basis \(\,B = \{v_1,\,v_2,\,\dots,\,v_n\}.\ \) If \(\,F,\,G\in\text{Hom}(V,W),\ \) then

\[F\,=\,G\quad\Leftrightarrow\quad F\,v_i\,=\,G\,v_i\,,\qquad i=1,2,\ldots,n.\]

Proof. \(\,\) The implication \(\ \Rightarrow\ \) being obvious, we shall prove the inference \(\ \Leftarrow\ \) only.

Let \(\ \displaystyle\,v\,=\,\sum_{i\,=\,1}^n\ a_i\,v_i\ \) be any vector in the space \(\,V.\ \) Then

\[F(v)\ =\ F\left(\,\sum_{i\,=\,1}^n\ a_i\,v_i\right)\ =\ \sum_{i\,=\,1}^n\ a_i\,Fv_i\ =\ \sum_{i\,=\,1}^n\ a_i\,Gv_i\ =\ G\left(\,\sum_{i\,=\,1}^n\ a_i\,v_i\right)\ =\ G(v)\,.\ \bullet\]

Corollary.

A linear transformation of a finite-dimensional vector space \(\,V\ \) into any vector space \(\,W\,\) (both over the same field \(\,K\)) is completely determined by its values on vectors of any basis of \(\,V.\)

\(\,\)

Now, we shall present a few examples of linear transformations.

Example 0. \(\,\) Let \(\ V\ \) and \(\ W\ \) be vector spaces over the field \(\,K.\ \) The following mappings are linear transformations:

1. The zero map \(\ \,\Theta:\,V\rightarrow W:\qquad \Theta(v)\ =\ \theta_W\,,\quad\forall\ v\in V.\)

2. The identity map \(\ \,I:\,V\rightarrow V:\qquad I(v)\ =\ v\,,\quad\forall\ v\in V.\)

3. The map \(\ \,F_a:\,V\rightarrow V:\qquad F_a(v)\ =\ a\,v\,,\quad a\in K,\ \forall\ v\in V.\)

The transformations \(\,I\ \) and \(\,F_a\ \) (for \(\,a\ne 0\)) \(\,\) are automorphisms of the space \(\,V.\)

Example 1. \(\,\) The following maps \(\ \,F:\,K^n\rightarrow K^m\ \) are linear transformations

1. \(\quad F \left[\begin{array}{l} x_1 \\ \dots \\ x_m \\ x_{m+1} \\ \dots \\ x_n \end{array}\right] \ =\ \left[\begin{array}{l} x_1 \\ \dots \\ x_m \end{array}\right]\,,\qquad (n\geq m).\)

   In particular, for \(\,m=1,\ \) the transformation \(\ F\ \) is a linear functional.

2. \(\quad F(\boldsymbol{x})\ =\ \boldsymbol{A}\,\boldsymbol{x}\,,\qquad\) where \(\ \boldsymbol{A}\in M_{m\times n}(K)\ \) is a given matrix, \(\ \,\boldsymbol{x}\in K^n.\)

Example 2. \(\,\) The matrix transpose operation

\[T(\boldsymbol{A})\ =\ \boldsymbol{A}^{\,T}\,,\qquad\boldsymbol{A}\in M_{m\times n}(K)\]

defines the linear transformation \(\ T:\,M_{m\times n}(K)\rightarrow M_{n\times m}(K).\)

Example 3. \(\\\) Given the matrices \(\ \boldsymbol{B}\in M_{k\times m}(K)\ \) and \(\ \boldsymbol{C}\in M_{n\times l}(K),\ \) due to the properties of matrix multiplication, the mapping \(\ F:\,M_{m\times n}(K)\rightarrow M_{k\times l}(K)\ \) defined as

\[F(\boldsymbol{A})\ =\ \boldsymbol{B}\boldsymbol{A}\boldsymbol{C}\,, \qquad\forall\ \boldsymbol{A}\in M_{m\times n}(K)\,,\]

is a linear transformation.

Example 4. \(\,\) Let \(\ V\ \) be the 3-dimensional space of geometrical vectors, \(\ \vec{a}\in V.\)

1. The map \(\ F:\,V\rightarrow V\ \) given by the cross product

   \[F(\vec{r})\ =\ \vec{a}\times\vec{r}\,,\qquad\forall\ \vec{r}\in V\,,\]

   is a (non-surjective) endomorphism of the space \(\ V.\)

2. The map \(\ f:\,V\rightarrow R\ \) given by the dot product

   \[F(\vec{r})\ =\ \vec{a}\cdot\vec{r}\,,\qquad\forall\ \vec{r}\in V\,,\]

   is a linear functional defined on the space \(\ V.\)

On the other hand, for a given number \(\,0\ne a\in R\ \) the transformation \(\ F:\,V\rightarrow V\ \)

\[F(\vec{r})\ =\ a\,\vec{r}\,,\qquad\forall\ \vec{r}\in V\,,\]

is an automorphism of the space \(\,V\ \) (see Example 0., item 3.).

Example 5. \(\,\) The mapping \(\,f:\,K^n\rightarrow K\ \) given by

\[\begin{split}f\left[\begin{array}{l} x_1 \\ x_2 \\ \dots \\ x_n \end{array}\right]\ \,:\,=\ \, x_1 + x_2 + \ \dots\ + x_n\end{split}\]

is a linear functional.

Example 6. \(\,\) We define the \(\,\) trace \(\,\) of a square matrix \(\ \boldsymbol{A}=[a_{ij}]_{n\times n}\in M_n(K)\ \) as the sum of its diagonal elements:

\[\text{Tr}\,\boldsymbol{A}\ \,:\,=\ \,\sum_{i\,=\,1}^n\ a_{ii}\,.\]

The properties of the matrix operations imply that the map \(\ \text{Tr}:\,M_n(K)\rightarrow K\ \,\) is a linear functional defined upon the algebra of square matrices of size \(\,n\,\) over the field \(\,K.\)

Example 7. \(\,\) Let \(\,\mathcal{C}_{[\,0,1\,]}^{\,\infty}\ \) denote the vector space of real functions defined on the interval \(\,[\,0,\,1\,]\ \) and having derivatives of any order. The map which ascribes to each function from \(\,\mathcal{C}_{[\,0,1\,]}^{\,\infty}\ \) its first derivative, is an endomorphism, since the operation of differentiating is linear.