Matrix Representation of Linear Transformations

Before we develop a general theory, we employ a simple example to present a connection between linear transformations and matrices.

Introduction

Consider a mapping \(\ F:\,R^3\rightarrow R^2\ \) given by the formula

(1)\[\begin{split}F\left[\begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array}\right]\ :\,=\ \left[\begin{array}{c} 2\,a_1+\,a_2-\,a_3 \\ 4\,a_1-\,2\,a_2+\,4\,a_3 \end{array}\right]\,.\end{split}\]

To see that \(\,F\,\) is a linear transformation, one can write the right hand side of the equation (1) as a product of two matrices:

\[\begin{split}F\left[\begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array}\right]\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \left[\begin{array}{c} a_1 \\ a_2 \\ a_3 \end{array}\right]\,.\end{split}\]

Now additivity and homogenity of the mapping \(\,F\,\) follows from the properties of matrix operations.

In this natural way, we associated the mapping \(\,F\in\text{Hom}(R^3,R^2)\ \) with the matrix

\[\begin{split}M(F)\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \in M_{2\times 3}(R)\,.\end{split}\]

Thanks to this matrix, the problem of determination of the image of a vector \(\,\boldsymbol{x}\in R^3\ \) under transformation \(\,F\,\) boils down to matrix multiplication:

\[F(\boldsymbol{x})\ =\ M(F)\cdot \boldsymbol{x}\,,\qquad \boldsymbol{x}\in R^3\,.\]

Let \(\ \boldsymbol{e}_1,\,\boldsymbol{e}_2,\,\boldsymbol{e}_3\ \) be vectors from the canonical basis of the space \(\,R^3.\ \) Note that \(\\\)

\[ \begin{align}\begin{aligned}\begin{split}\begin{array}{l} F\boldsymbol{e}_1\ =\ F \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right]\ =\ \left[\begin{array}{c} 2 \\ 4 \end{array}\right]\,, \\ \\ F\boldsymbol{e}_2\ =\ F \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]\ =\ \left[\begin{array}{r} 1 \\ -2 \end{array}\right]\,, \\ \\ F\boldsymbol{e}_3\ =\ F \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right]\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right]\ =\ \left[\begin{array}{r} -1 \\ 4 \end{array}\right]\,. \end{array}\end{split}\\\;\end{aligned}\end{align} \]

As one can see, the matrix \(\ M(F)\ \) consists of columns which are the images of suitable vectors of the canonical basis of the space \(\ R^3:\ \) \(\ M(F)\ =\ [\,F\boldsymbol{e}_1\,|\,F\boldsymbol{e}_2\,|\,F\boldsymbol{e}_3\,]\,.\)

More generally, one can associate a linear mapping \(\,F\in\text{Hom}(K^n,K^m)\ \) with the matrix whose \(\,j\)-th column is the image of the \(\,j\)-th vector from the canonical basis of the space \(\ K^n\,,\ \ j=1,2,\dots,n.\ \)

Such defined mapping \(\,M\,\) from the space of linear transformations \(\ \text{Hom}(K^n,K^m)\ \) into the space \(\ M_{m\times n}(K)\ \) of rectangular matrices may be written as follows:

(2)\[M:\quad \text{Hom}(K^n,K^m)\,\ni\,F \ \ \rightarrow\ \ M(F)\,:\,=\,[\,F\boldsymbol{e}_1\,|\,\dots\,|\,F\boldsymbol{e}_n\,]\,\in\,M_{m\times n}(K)\,,\]

where \(\ \mathcal{E}=(\boldsymbol{e}_1,\,\dots,\,\boldsymbol{e}_n)\ \,\) denotes the canonical basis of the space \(\,K^n.\ \) Then the image of any vector \(\,\boldsymbol{x}\in K^n\ \) may be obtained by multiplication of this vector (on the left hand side) by the matrix \(\,M(F):\)

\[\boldsymbol{y}\,=\,F(\boldsymbol{x})\quad\Rightarrow\quad \boldsymbol{y}\ =\ M(F)\,\cdot\,\boldsymbol{x}\,,\qquad \boldsymbol{x}\in K^n\,,\ \ \boldsymbol{y}\in K^m\,.\]

We will generalise this further and define a matrix of linear transformation \(\ F:V\rightarrow W,\ \) where \(\ V\ \) and \(\ W\ \) are \(\,\) arbitrary \(\,\) finite dimensional vector spaces over a field \(\ K\,,\ \) each with a chosen basis.

Matrix of a Linear Transformation

Consider two finite dimensional vector spaces over a field \(\,K:\ \\\) \(n\)-dimensional space \(\,V\,\) with a basis \(\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\,,\ \\\) \(m\)-dimensional space \(\,W\,\) with a basis \(\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\ \\\) and a linear transformation \(\,F\in\text{Hom}(V,W)\,.\)

Images of the basis vectors from \(\ \mathcal{B}\ \) belong to the space \(\,W,\ \) and so may be written as linear combinations of vectors from the basis \(\ \mathcal{C}:\)

(3)\[\begin{split}\begin{array}{l} Fv_1\ =\ f_{11}\,w_1\,+\ f_{21}\,w_2\,+\ \dots\ +\ f_{m1}\,w_m \\ Fv_2\ =\ f_{12}\,w_1\,+\ f_{22}\,w_2\,+\ \dots\ +\ f_{m2}\,w_m \\ \dots \\ Fv_n\ =\ f_{1n}\,w_1\,+\ f_{2n}\,w_2\,+\ \dots\ +\ f_{mn}\,w_m \end{array}\end{split}\]

A matrix \(\ \boldsymbol{F}=[\,f_{ij}\,]_{m\times n}(K)\ \) obtained in such a way is \(\,\) by definition \(\,\) a matrix \(\,M_{\mathcal{B}\mathcal{C}}(F)\ \) of a linear transformation \(\ F\ \) in bases \(\ \mathcal{B}\ \,\) and \(\, \ \mathcal{C}:\)

\[\begin{split}M_{\mathcal{B}\mathcal{C}}(F)\ :\,=\ \left[ \begin{array}{cccc} f_{11} & f_{12} & \dots & f_{1n} \\ f_{21} & f_{22} & \dots & f_{2n} \\ \dots & \dots & \dots & \dots \\ f_{m1} & f_{m2} & \dots & f_{mn} \end{array} \right]\,.\end{split}\]

Moreover, the entries \(\ f_{1j},\,f_{2j},\,\dots,\,f_{mj}\,\ \) from the \(\,j\)-th column of the matrix \(\\\) are coordinates of the vector \(\ Fv_j\ \) in the basis \(\ \mathcal{C},\ \ j=1,2,\dots,n.\ \)

Definition. \(\\\)

Let \(\ \,V\ \,\) and \(\, \ W\ \,\) be two finte dimensional vector spaces over a field \(\,K,\ \) \(\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\ \) a basis of the space \(\ \,V,\ \) and \(\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,\) a basis of the space \(\ W.\ \,\) Then the \(\ j\)-th column of the matrix \(\ M_{\mathcal{B}\mathcal{C}}(F)\ \) of a linear transformation \(\,F\in\text{Hom}(V,W)\ \) in bases \(\ \mathcal{B}\ \) and \(\ \mathcal{C}\ \) is a column of coordinates \(\,\) (in the basis \(\ \mathcal{C}\,\)) \(\,\) of the image \(\,\) - \(\,\) under the transformation \(\,F\ \) \(\,\) - \(\,\) of the \(\ j\)-th vector from the basis \(\ \mathcal{B}\quad (j=1,2,\dots,n).\)

Hence, \(\ \,M_{\mathcal{B}\mathcal{C}}(F)\ =\ \,[\,f_{ij}\,]_{m\times n}\,,\ \,\) where the entries \(\ f_{ij}\ \) are defined by relations

\[Fv_j\;=\ \sum_{i\,=\,1}^m\ f_{ij}\ w_i\,,\qquad j=1,2,\dots,n\,.\]

Example.

We discuss an operation of differentiation defined on a set of real polynomials.

Let \(\,V\ \) be a vector space of polynomials in one variable \(\,x\ \) of degree (not greater than) \(\,n,\ \,\) and \(\ \,W\ \ \) a space of such polynomials of degree (not greater than) \(\ n-1:\)

\[ \begin{align}\begin{aligned}V\ =\ \{\,a_0\,+\,a_1\,x\,+\,a_2\,x^2\,+\,a_3\,x^3\,+\,\ldots\,+\,a_n\,x^n: \quad a_i\in R\,,\quad i=0,1,\dots,n\,\}\,,\\ W\ =\ \{\,b_0\,+\,b_1\,x\,+\,b_2\,x^2\,+\,\ldots\,+\,b_{n-1}\,x^{n-1}: \quad b_i\in R\,,\quad i=0,1,\dots,n-1\,\}\,.\end{aligned}\end{align} \]

\(\dim\,V=\,n+1\,,\ \ \mathcal{B}\,=\,(1,\,x,\,x^2,\,x^3,\,\dots,\,x^n)\,;\quad \dim\,W=\,n\,,\ \ \mathcal{C}\,=\,(1,\,x,\,x^2,\,\dots,\,x^{n-1})\,.\)

A differential operator \(\ D\equiv {d\over dx}\ \) transforms the space \(\,V\ \) linearly into the space \(\,W.\) To determine a matrix of this operation in bases \(\,\mathcal{B}\,\) and \(\,\mathcal{C} ,\,\) we write decompositions (3) of images of the consecutive vectors from the basis \(\,\mathcal{B}\,\) in the basis \(\, \mathcal{C}:\)

\begin{alignat*}{7} D\,1\:\ & {\,} = {\,} & 0 & {\quad} = {\quad} & 0\cdot 1 & {\ } + {\ } & 0\cdot x & {\ } + {\ } & 0\cdot x^2 & {\ } + {\ } & \dots & {\ } + {\ } & 0\cdot x^{n-1} \\ D\,x\,\ & {\,} = {\,} & 1 & {\quad} = {\quad} & 1\cdot 1 & {\ } + {\ } & 0\cdot x & {\ } + {\ } & 0\cdot x^2 & {\ } + {\ } & \dots & {\ } + {\ } & 0\cdot x^{n-1} \\ D\,x^2 & {\,} = {\,} & 2\,x & {\quad} = {\quad} & 0\cdot 1 & {\ } + {\ } & 2\cdot x & {\ } + {\ } & 0\cdot x^2 & {\ } + {\ } & \dots & {\ } + {\ } & 0\cdot x^{n-1} \\ D\,x^3 & {\,} = {\,} & 3\,x^2 & {\quad} = {\quad} & 0\cdot 1 & {\ } + {\ } & 0\cdot x & {\ } + {\ } & 3\cdot x^2 & {\ } + {\ } & \dots & {\ } + {\ } & 0\cdot x^{n-1} \\ \dots & {\,} {\,} & \dots & {\quad} {\quad} & \dots & {\ } {\ } & \dots & {\ } {\ } & \dots & {\ } {\ } & \dots & {\ } {\ } & \dots \\ D\,x^n & {\,} = {\,} & n\,x^{n-1} & {\quad} = {\quad} & 0\cdot 1 & {\ } + {\ } & 0\cdot x & {\ } + {\ } & 0\cdot x^2 & {\ } + {\ } & \dots & {\ } + {\ } & n\cdot x^{n-1} \end{alignat*}

(4)\[ \begin{align}\begin{aligned}\begin{split}M_{\mathcal{B}\mathcal{C}}(D)\ =\ \left[ \begin{array}{cccccc} 0 & 1 & 0 & 0 & \dots & 0 \\ 0 & 0 & 2 & 0 & \dots & 0 \\ 0 & 0 & 0 & 3 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & 0 & \dots & n \end{array} \right]\ \in\,M_{n\times (n+1)}(R)\,.\end{split}\\\;\end{aligned}\end{align} \]

We introduce further notation in order to write clearly a matrix \(\,M_{\mathcal{B}\mathcal{C}}(F)\,\) in a column form. Corollary to Theorem 8. implies that \(\,n\)-dimensional space \(\,V\ \) is isomorphic to the space \(\,K^n,\,\) and \(\, m\)-dimensional space \(\,W\ \) is isomorphic to the space \(\ K^m:\quad V\,\simeq\,K^n\,,\qquad W\,\simeq\,K^m\,.\)

For the spaces \(\,V\,\) and \(\, W\ \) we fixed the bases

\[\mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n) \qquad\text{and}\qquad \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,.\]

Let

\[\mathcal{E}\,=\,(e_1,\,e_2,\,\dots,\,e_n) \qquad\text{and}\qquad \mathcal{F}\,=\,(f_1,\,f_2,\,\dots,\,f_m)\]

be the canonical bases of the spaces \(\,K^n\ \,\) and \(\, K^m.\)

Then the mappings \(\ I_{\mathcal{B}}:\,V\rightarrow K^n \,\) and \(\, I_{\mathcal{C}}:\,W\rightarrow K^m\,\) defined by fixing the images on the basis vectors (for the basis \(\,\mathcal{B}\ \) or \(\ \mathcal{C}\,\) respectively):

\[I_{\mathcal{B}}(v_j)\ :\,=\ e_j\,,\quad j=1,2,\dots,n\,, \qquad I_{\mathcal{C}}(w_i)\ :\,=\ f_i\,,\quad i=1,2,\dots,m\,,\]

are examples of isomorphisms: \(\ I_{\mathcal{B}}\in\text{Iso}(V,K^n)\,,\ \,I_{\mathcal{C}}\in\text{Iso}(W,K^m)\,.\)

For any vectors \(\displaystyle\quad v\,=\,\sum_{j\,=\,1}^n\ a_j\,v_j\,\in V\,,\quad w\,=\,\sum_{i\,=\,1}^m\ b_i\,w_i\,\in W\,:\)

\[ \begin{align}\begin{aligned}\begin{split}I_{\mathcal{B}}(v)\ =\ I_{\mathcal{B}}\,\left(\,\sum_{j\,=\,1}^n\ a_j\,v_j\right)\ =\ \sum_{j\,=\,1}^n\ a_j\,I_{\mathcal{B}}(v_j)\ =\ \sum_{j\,=\,1}^n\ a_j\,e_j\ =\ \left[\begin{array}{c} a_1 \\ a_2 \\ \dots \\ a_n \end{array}\right]\,,\end{split}\\\begin{split}I_{\mathcal{C}}(w)\ =\ I_{\mathcal{C}}\,\left(\,\sum_{i\,=\,1}^m\ b_i\,w_i\right)\ =\ \sum_{i\,=\,1}^m\ b_i\,I_{\mathcal{C}}(w_i)\ =\ \sum_{i\,=\,1}^m\ b_i\,f_i\ =\ \left[\begin{array}{c} b_1 \\ b_2 \\ \dots \\ b_m \end{array}\right]\,.\end{split}\end{aligned}\end{align} \]

Hence, the isomorphism \(\ I_{\mathcal{B}}\ \) transforms a vector \(\,v\in V\ \) into a column of the coordinates of this vector in a basis \(\ \mathcal{B},\ \,\) and \(\,\) the isomorphism \(\ \,I_{\mathcal{C}}\ \) transforms a vector \(\,w\in W\ \) into a column of the coordinates of this vector in a basis \(\ \mathcal{C}.\ \) A matrix of the linear transformation \(\ F\in\text{Hom}(V,W)\ \) in bases \(\ \mathcal{B}\ \,\) and \(\,\mathcal{C}\ \) may be now written in a column form

\[M_{\mathcal{B}\mathcal{C}}(F)\ \,=\ \, \left[\;I_{\mathcal{C}}(Fv_1\,|\,I_{\mathcal{C}}(Fv_2\,|\ \dots\ |\, I_{\mathcal{C}}(Fv_n\,\right]\,.\]

Basic Theorems

The purpose of introducing matrix representation of linear transformations explains

Theorem 10. \(\\\)

Let \(\ F\in\text{Hom}(V,W),\ \) where \(\,V \,\) and \(\, W\,\) are vector spaces over a field \(\,K\,\) with bases \(\ \mathcal{B}\ \,\) and \(\ \mathcal{C}.\ \) If a vector \(\,w\in W\,\) is an image of a vector \(\,v\in V\,\) under the transformation \(\,F, \,\) then the column of coordinates (in a basis \(\,\mathcal{C}\,\)) of the vector \(\ w\ \) is equal to a product of the transformation matrix of \(\,F\,\) in bases \(\, \mathcal{B}\,\) and \(\,\mathcal{C}\,\) and a column of coordinates (in a basis \(\,\mathcal{B}\,\)) \(\,\) of the vector \(\,v:\)

(5)\[w\,=\,F(v)\qquad\Rightarrow\qquad I_{\mathcal{C}}(w)\ =\ M_{\mathcal{B}\mathcal{C}}(F)\,\cdot\,I_{\mathcal{B}}(v)\,.\]

In this way, an abstract issue of finding an image of a vector \(\,v\ \) under a transformation \(\,F\ \) boils down to concrete calculation on matrices.

Proof. \(\,\) We keep the above notation:

\[ \begin{align}\begin{aligned}\mathcal{B}\,=\,(v_1,\,v_2,\,\dots,\,v_n)\,,\qquad\mathcal{C}\,=\,(w_1,\,w_2,\,\dots,\,w_m)\,,\\v\,=\,\sum_{j\,=\,1}^n\ a_j\,v_j\,,\quad w\,=\,\sum_{i\,=\,1}^m\ b_i\,w_i\,,\quad M_{\mathcal{B}\mathcal{C}}(F)\,=\,[\,f_{ij}\,]_{m\times n}\,.\end{aligned}\end{align} \]

Then

\[ \begin{align}\begin{aligned}w\ =\ F(v)\ =\ F\,\left(\,\sum_{j\,=\,1}^n\ a_j\,v_j\right)\ \ =\ \ \sum_{j\,=\,1}^n\ a_j\,F(v_j)\ \ =\\=\ \ \sum_{j\,=\,1}^n\ a_j\,\left(\,\sum_{i\,=\,1}^m\ f_{ij}\ w_i\right)\ \ =\ \ \sum_{i\,=\,1}^m\,\left(\,\sum_{j\,=\,1}^n\ f_{ij}\ a_j\right)\ w_i\,.\end{aligned}\end{align} \]

By uniqueness of representation of a vector \(\,w\ \) in the basis \(\,\mathcal{C},\)

(6)\[b_i\ =\ \sum_{j\,=\,1}^n\ f_{ij}\ a_j\,,\qquad i=1,2,\dots,m\,.\]

The relations (6) describe equality of matrices \(\\\)

\[ \begin{align}\begin{aligned}\begin{split}\left[\begin{array}{c} b_1 \\ b_2 \\ \dots \\ b_m \end{array}\right]\ =\ \left[\begin{array}{cccc} f_{11} & f_{12} & \dots & f_{1n} \\ f_{21} & f_{22} & \dots & f_{2n} \\ \dots & \dots & \dots & \dots \\ f_{m1} & f_{m2} & \dots & f_{mn} \end{array} \right] \left[\begin{array}{c} a_1 \\ a_2 \\ \dots \\ a_n \end{array}\right]\,,\end{split}\\\;\\\text{that is}\qquad I_{\mathcal{C}}(w)\ =\ M_{\mathcal{B}\mathcal{C}}(F)\,\cdot\,I_{\mathcal{B}}(v)\,.\end{aligned}\end{align} \]

Example.

Let us come back to a differential operator \(\ D = {d\over dx}\ \,\) viewed as a linear transformation of the space \(\,V\ \) of real polynomials of degree \(\,n\ \) into the space \(\,W\ \) of polynomials of degree \(\,n-1.\ \) The matrix associated with this operation in natural bases of spaces \(\ V\ \,\) and \(\, W\ \) is given by (4).

If \(\ v\,=\,a_0\,+\,a_1\,x\,+\,a_2\,x^2\,+\,a_3\,x^3\,+\,\ldots\,+\,a_n\,x^n\,\in V,\)

then \(\quad w\,\equiv D(v)\,=\,a_1\,+\,2\,a_2\,x\,+\,3\,a_3\,x^2\ +\ \ldots\ +n\,a_n\,x^{n-1}\,.\)

Matrix relation between the coordinates of the polynomials \(\,v\ \,\) and \(\, w:\)

\[\begin{split}\left[ \begin{array}{c} a_1 \\ 2\,a_2 \\ 3\,a_3 \\ \dots \\ n\,a_n \end{array} \right]\ \ =\ \ \left[ \begin{array}{cccccc} 0 & 1 & 0 & 0 & \dots & 0 \\ 0 & 0 & 2 & 0 & \dots & 0 \\ 0 & 0 & 0 & 3 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & 0 & \dots & n \end{array} \right]\ \left[ \begin{array}{c} a_0 \\ a_1 \\ a_2 \\ a_3 \\ \dots \\ a_n \end{array} \right]\end{split}\]

is precisely the relation (5) in Theorem 10.

\(\;\)

Let us explain nature of the relation between linear transformations and matrices more precisely. So far we considered the following vector spaces (all of them over the same field \(\,K\,\)):

• \(n\)-dimensional space \(\,V\,\) with basis \(\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\,,\ \)

  \(m\)-dimensional space \(\,W\,\) with basis \(\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,;\)
• space \(\ \text{Hom}(V,W)\ \) of linear transformations of the space \(\ V\ \) into the space \(\ W;\)
• space \(\ M_{m\times n}(K)\ \) of rectangular matrices with the entries from the field \(\ K. \,\)

\(\;\)

Theorem 11. \(\\\)

The mapping

\[M_{\mathcal{B}\mathcal{C}}:\quad \text{Hom}(V,W)\ni F\ \rightarrow\ M_{\mathcal{B}\mathcal{C}}(F):\,= \left[\;I_{\mathcal{C}}(Fv_1\,|\,\dots\,|\, I_{\mathcal{C}}(Fv_n\,\right]\in M_{m\times n}(K)\]

is an isomorphism of the vector spaces \(\ \text{Hom}(V,W)\,\) and \(\, M_{m\times n}(K).\)

\(\;\)

Proof is preceded with a reminder of definitions of operations on linear transformations which make \(\,\text{Hom}(V,W)\,\) a vector space. If \(\,F_1,F_2,F\in\text{Hom}(V,W),\ a\in K,\,\) then

\begin{eqnarray*} (F_1+F_2)(v) & :\;= & F_1(v)\,+\,F_2(v) \\ (a\,F)(v) & :\;= & a\cdot F(v)\,,\qquad v\in V\,. \end{eqnarray*}

To show that \(\,M_{\mathcal{B}\mathcal{C}}\ \) is an isomorphism, we have to prove its additivity, homogenity and bijectivity.

1. Additivity. \(\,\)

   Let \(\,F_1,F_2\,\in\,\text{Hom}(V,W).\ \) Then he \(\,j\)-th column of the matrix \(\,M_{\mathcal{B}\mathcal{C}}(F_1+F_2)\)

   \[I_{\mathcal{C}}\,[\,(F_1+F_2)(v_j)\,]\ =\ I_{\mathcal{C}}\,[\,F_1(v_j)+F_2(v_j)\,]\ =\ I_{\mathcal{C}}\,[\,F_1(v_j)\,]+I_{\mathcal{C}}\,[\,F_2(v_j)\,]\]

   is a sum of the \(\,j\)-th columns of the matrices \(\ M_{\mathcal{B}\mathcal{C}}(F_1)\ \) and \(\ \,M_{\mathcal{B}\mathcal{C}}(F_2)\,,\ \ j=1,2,\dots,n.\ \,\) Hence,

   \[M_{\mathcal{B}\mathcal{C}}(F_1+F_2)\ =\ M_{\mathcal{B}\mathcal{C}}(F_1) \,+\,M_{\mathcal{B}\mathcal{C}}(F_2)\,.\]

2. Homogenity.

   Let \(\,F\in\text{Hom}(V,W),\ \ a\in K.\ \,\) Then the \(\,j\)-th column of the matrix \(\,M_{\mathcal{B}\mathcal{C}}(aF)\)

   \[I_{\mathcal{C}}\,[\,(aF)(v_j)\,]\ =\ I_{\mathcal{C}}\,[\,a\cdot F(v_j)\,]\ =\ a\cdot I_{\mathcal{C}}\,[\,F(v_j)\,]\]

   is the \(\, j\)-th column of the matrix \(\,M_{\mathcal{B}\mathcal{C}}(F)\,,\ \ j=1,2,\dots,n, \,\) multiplied by \(\,a.\,\) Hence,

   \[M_{\mathcal{B}\mathcal{C}}(a\,F)\ =\ a\,M_{\mathcal{B}\mathcal{C}}(F)\,.\]

3. Bijectivity.

   We have to show that every matrix \(\,\boldsymbol{F}\in M_{m\times n}(K)\,\) is associated with exactly one mapping \(\,F\in\text{Hom}(V,W).\,\) Indeed, columns of the matrix \(\boldsymbol{F}\,\) determine (by the coordinates in the basis \(\, \mathcal{C}\,\)) \(\,\) images \(\, Fv_j\,\) of basis vectors \(\,v_j\in\mathcal{B},\,\) and thus (cf. Corollary to Theorem 5.) \(\,\) the transformation \(\ F\ \) is uniquely defined.

\(\;\)

On the basis of Theorem 8. we may now write

Corollary.

If \(\,V\ \,\) and \(\, W\ \) are finite dimensional vector spaces over a field \(\,K,\ \,\) then

\[\dim\,\text{Hom}(V,W)\ =\ \dim\,V\,\cdot\,\dim\,W\,.\]

\(\;\)

We consider one more case: \(\,V=K^n\,\) with the canonical basis \(\,\mathcal{E}=(\boldsymbol{e}_1,\boldsymbol{e}_2,\dots,\boldsymbol{e}_n)\,,\) \(\,W=K^m\,\) with the canonical basis \(\, \mathcal{F}=(\boldsymbol{f}_1,\boldsymbol{f}_2,\dots,\boldsymbol{f}_m),\,\) and \(\, F\in\text{Hom}(K^n,K^m).\)

A matrix of the transformation \(\,F\,\) in the canonical bases \(\, \mathcal{E}\,\) and \(\, \mathcal{F}\,\) is of the form

\[M_{\mathcal{E}\mathcal{F}}(F)\ =\ [\,I_{\mathcal{F}}(F\boldsymbol{e}_1)\,|\,I_{\mathcal{F}}(F\boldsymbol{e}_2)\,|\,\dots\, |\,I_{\mathcal{F}}(F\boldsymbol{e}_n)\,]\,.\]

However, in the space \(\,K^m\ \) each vector is a column of its coordinates in the canonical basis: \(\ \ I_{\mathcal{F}}(\boldsymbol{w})=\boldsymbol{w},\ \ \boldsymbol{w}\in K^m.\ \) If we denote the matrix of the transformation \(\,F\ \) in the canonical basis simply by \(\,M(F),\ \) we obtain a simplified formula:

\[M(F)\ =\ [\,F\boldsymbol{e}_1\,|\,F\boldsymbol{e}_2\,|\,\dots\,|\,F\boldsymbol{e}_n\,]\,,\]

which has been introduced earlier in the equation (2). The formula (5) in Theorem 10. takes now the form

\[\boldsymbol{y}\,=\,F(\boldsymbol{x})\quad\Rightarrow\quad \boldsymbol{y}\ =\ M(F)\,\cdot\,\boldsymbol{x}\,,\qquad \boldsymbol{x}\in K^n\,,\ \ \boldsymbol{y}\in K^m\,.\]