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 :math:`\ F:\,R^3\rightarrow R^2\ ` given by the formula .. math:: :label: ex_0 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]\,. To see that :math:`\,F\,` is a linear transformation, one can write the right hand side of the equation :eq:`ex_0` as a product of two matrices: .. math:: 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]\,. Now additivity and homogenity of the mapping :math:`\,F\,` follows from the properties of matrix operations. In this natural way, we associated the mapping :math:`\,F\in\text{Hom}(R^3,R^2)\ ` with the matrix .. math:: M(F)\ =\ \left[\begin{array}{rrr} 2 & 1 & -1 \\ 4 & -2 & 4 \end{array}\right] \in M_{2\times 3}(R)\,. Thanks to this matrix, the problem of determination of the image of a vector :math:`\,\boldsymbol{x}\in R^3\ ` under transformation :math:`\,F\,` boils down to matrix multiplication: .. math:: F(\boldsymbol{x})\ =\ M(F)\cdot \boldsymbol{x}\,,\qquad \boldsymbol{x}\in R^3\,. Let :math:`\ \boldsymbol{e}_1,\,\boldsymbol{e}_2,\,\boldsymbol{e}_3\ ` be vectors from the canonical basis of the space :math:`\,R^3.\ ` Note that :math:`\\` .. math:: \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} \; As one can see, the matrix :math:`\ M(F)\ ` consists of columns which are the images of suitable vectors of the canonical basis of the space :math:`\ R^3:\ ` :math:`\ M(F)\ =\ [\,F\boldsymbol{e}_1\,|\,F\boldsymbol{e}_2\,|\,F\boldsymbol{e}_3\,]\,.` .. Uogólnienie tego przykładu opiera się na stwierdzeniu, że każde przekształcenie liniowe przestrzeni :math:`\,K^n\ ` w przestrzeń :math:`\,K^m\ ` ma postać :eq:`ex_0`, to znaczy współrzędne obrazu są jednorodnymi liniowymi funkcjami współrzędnych argumentu. More generally, one can associate a linear mapping :math:`\,F\in\text{Hom}(K^n,K^m)\ ` with the matrix whose :math:`\,j`-th column is the image of the :math:`\,j`-th vector from the canonical basis of the space :math:`\ K^n\,,\ \ j=1,2,\dots,n.\ ` Such defined mapping :math:`\,M\,` from the space of linear transformations :math:`\ \text{Hom}(K^n,K^m)\ ` into the space :math:`\ M_{m\times n}(K)\ ` of rectangular matrices may be written as follows: .. math:: :label: intro 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 :math:`\ \mathcal{E}=(\boldsymbol{e}_1,\,\dots,\,\boldsymbol{e}_n)\ \,` denotes the canonical basis of the space :math:`\,K^n.\ ` Then the image of any vector :math:`\,\boldsymbol{x}\in K^n\ ` may be obtained by multiplication of this vector (on the left hand side) by the matrix :math:`\,M(F):` .. math:: \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\,. .. W następnym uogólnieniu pokażemy, :math:`\,` jak przekształceniu liniowemu *dowolnych* skończenie wymiarowych przestrzeni nad ciałem :math:`\,K,\ ` w których wybrano bazy, można przyporządkować macierz o elementach z :math:`\,K.` We will generalise this further and define a matrix of linear transformation :math:`\ F:V\rightarrow W,\ ` where :math:`\ V\ ` and :math:`\ W\ ` are :math:`\,` *arbitrary* :math:`\,` finite dimensional vector spaces over a field :math:`\ K\,,\ ` each with a chosen basis. .. _`matrix_of_lin_trans`: Matrix of a Linear Transformation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Consider two finite dimensional vector spaces over a field :math:`\,K:\ \\` :math:`n`-dimensional space :math:`\,V\,` with a basis :math:`\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\,,\ \\` :math:`m`-dimensional space :math:`\,W\,` with a basis :math:`\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\ \\` and a linear transformation :math:`\,F\in\text{Hom}(V,W)\,.` Images of the basis vectors from :math:`\ \mathcal{B}\ ` belong to the space :math:`\,W,\ ` and so may be written as linear combinations of vectors from the basis :math:`\ \mathcal{C}:` .. .. math:: :label: exps \begin{array}{l} Fv_1\ =\ a_{11}\,w_1\,+\ a_{21}\,w_2\,+\ \dots\ +\ a_{m1}\,w_m \\ Fv_2\ =\ a_{12}\,w_1\,+\ a_{22}\,w_2\,+\ \dots\ +\ a_{m2}\,w_m \\ \dots \\ Fv_n\ =\ a_{1n}\,w_1\,+\ a_{2n}\,w_2\,+\ \dots\ +\ a_{mn}\,w_m \end{array} .. math:: :label: exps \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} A matrix :math:`\ \boldsymbol{F}=[\,f_{ij}\,]_{m\times n}(K)\ ` obtained in such a way is :math:`\,` *by definition* :math:`\,` a matrix :math:`\,M_{\mathcal{B}\mathcal{C}}(F)\ ` of a linear transformation :math:`\ F\ ` in bases :math:`\ \mathcal{B}\ \,` and :math:`\, \ \mathcal{C}:` .. .. math:: M_{\mathcal{B}\mathcal{C}}(F)\ :\,=\ \left[ \begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \dots & \dots & \dots & \dots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{array} \right]\,. .. math:: 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]\,. Moreover, the entries :math:`\ f_{1j},\,f_{2j},\,\dots,\,f_{mj}\,\ ` from the :math:`\,j`-th column of the matrix :math:`\\` are coordinates of the vector :math:`\ Fv_j\ ` in the basis :math:`\ \mathcal{C},\ \ j=1,2,\dots,n.\ ` .. Wynika stąd następująca .. admonition:: Definition. :math:`\\` Let :math:`\ \,V\ \,` and :math:`\, \ W\ \,` be two finte dimensional vector spaces over a field :math:`\,K,\ ` :math:`\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\ ` a basis of the space :math:`\ \,V,\ ` and :math:`\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,` a basis of the space :math:`\ W.\ \,` Then the :math:`\ j`-th column of the matrix :math:`\ M_{\mathcal{B}\mathcal{C}}(F)\ ` of a linear transformation :math:`\,F\in\text{Hom}(V,W)\ ` in bases :math:`\ \mathcal{B}\ ` and :math:`\ \mathcal{C}\ ` is a column of coordinates :math:`\,` (in the basis :math:`\ \mathcal{C}\,`) :math:`\,` of the image :math:`\,` - :math:`\,` under the transformation :math:`\,F\ ` :math:`\,` - :math:`\,` of the :math:`\ j`-th vector from the basis :math:`\ \mathcal{B}\quad (j=1,2,\dots,n).` Hence, :math:`\ \,M_{\mathcal{B}\mathcal{C}}(F)\ =\ \,[\,f_{ij}\,]_{m\times n}\,,\ \,` where the entries :math:`\ f_{ij}\ ` are defined by relations .. math:: 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 :math:`\,V\ ` be a vector space of polynomials in one variable :math:`\,x\ ` of degree (not greater than) :math:`\,n,\ \,` and :math:`\ \,W\ \ ` a space of such polynomials of degree (not greater than) :math:`\ n-1:` .. math:: 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\,\}\,. :math:`\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})\,.` .. \begin{array}{lcl} \dim\,V\,=\,n+1\,, & \qquad & \text{baza:}\quad \mathcal{B}\,=\,(1,\,x,\,x^2,\,x^3,\,\dots,\,x^n)\,, \\ \dim\,w\,=\,n\,, & \qquad & \text{baza:}\quad \mathcal{C}\,=\,(1,\,x,\,x^2,\,\dots,\,x^{n-1})\,. \end{array} A differential operator :math:`\ D\equiv {d\over dx}\ ` transforms the space :math:`\,V\ ` linearly into the space :math:`\,W.` To determine a matrix of this operation in bases :math:`\,\mathcal{B}\,` and :math:`\,\mathcal{C} ,\,` we write decompositions :eq:`exps` of images of the consecutive vectors from the basis :math:`\,\mathcal{B}\,` in the basis :math:`\, \mathcal{C}:` .. math:: :nowrap: \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*} .. math:: :label: MBC_D 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)\,. \; We introduce further notation in order to write clearly a matrix :math:`\,M_{\mathcal{B}\mathcal{C}}(F)\,` in a column form. Corollary to Theorem 8. implies that :math:`\,n`-dimensional space :math:`\,V\ ` is isomorphic to the space :math:`\,K^n,\,` and :math:`\, m`-dimensional space :math:`\,W\ ` is isomorphic to the space :math:`\ K^m:\quad V\,\simeq\,K^n\,,\qquad W\,\simeq\,K^m\,.` .. .. math:: V\,\simeq\,K^n\,,\qquad W\,\simeq\,K^m\,. For the spaces :math:`\,V\,` and :math:`\, W\ ` we fixed the bases .. math:: \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n) \qquad\text{and}\qquad \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,. Let .. math:: \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 :math:`\,K^n\ \,` and :math:`\, K^m.` Then the mappings :math:`\ I_{\mathcal{B}}:\,V\rightarrow K^n \,` and :math:`\, I_{\mathcal{C}}:\,W\rightarrow K^m\,` defined by fixing the images on the basis vectors (for the basis :math:`\,\mathcal{B}\ ` or :math:`\ \mathcal{C}\,` respectively): .. określone wzorami .. math:: 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: :math:`\ I_{\mathcal{B}}\in\text{Iso}(V,K^n)\,,\ \,I_{\mathcal{C}}\in\text{Iso}(W,K^m)\,.` .. Odwzorowania :math:`\ I_{\mathcal{B}}\ \ \text{oraz}\ \ I_{\mathcal{C}}\ \,` zostały określone poprzez zadanie obrazów wektorów bazy, odpowiednio bazy :math:`\ \mathcal{B}\ \,` albo bazy :math:`\ \,\mathcal{C}.` For any vectors :math:`\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\,:` .. math:: 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]\,, 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]\,. Hence, the isomorphism :math:`\ I_{\mathcal{B}}\ ` transforms a vector :math:`\,v\in V\ ` into a column of the coordinates of this vector in a basis :math:`\ \mathcal{B},\ \,` and :math:`\,` the isomorphism :math:`\ \,I_{\mathcal{C}}\ ` transforms a vector :math:`\,w\in W\ ` into a column of the coordinates of this vector in a basis :math:`\ \mathcal{C}.\ ` A matrix of the linear transformation :math:`\ F\in\text{Hom}(V,W)\ ` in bases :math:`\ \mathcal{B}\ \,` and :math:`\,\mathcal{C}\ ` may be now written in a column form .. math:: 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 .. admonition:: Theorem 10. :math:`\\` Let :math:`\ F\in\text{Hom}(V,W),\ ` where :math:`\,V \,` and :math:`\, W\,` are vector spaces over a field :math:`\,K\,` with bases :math:`\ \mathcal{B}\ \,` and :math:`\ \mathcal{C}.\ ` If a vector :math:`\,w\in W\,` is an image of a vector :math:`\,v\in V\,` under the transformation :math:`\,F, \,` then the column of coordinates (in a basis :math:`\,\mathcal{C}\,`) of the vector :math:`\ w\ ` is equal to a product of the transformation matrix of :math:`\,F\,` in bases :math:`\, \mathcal{B}\,` and :math:`\,\mathcal{C}\,` and a column of coordinates (in a basis :math:`\,\mathcal{B}\,`) :math:`\,` of the vector :math:`\,v:` .. math:: :label: fund 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 :math:`\,v\ ` under a transformation :math:`\,F\ ` boils down to concrete calculation on matrices. **Proof.** :math:`\,` We keep the above notation: .. math:: \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}\,. Then .. math:: 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\,. By uniqueness of representation of a vector :math:`\,w\ ` in the basis :math:`\,\mathcal{C},` .. math:: :label: bfa b_i\ =\ \sum_{j\,=\,1}^n\ f_{ij}\ a_j\,,\qquad i=1,2,\dots,m\,. The relations :eq:`bfa` describe equality of matrices :math:`\\` .. math:: \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]\,, \; \text{that is}\qquad I_{\mathcal{C}}(w)\ =\ M_{\mathcal{B}\mathcal{C}}(F)\,\cdot\,I_{\mathcal{B}}(v)\,. **Example.** Let us come back to a differential operator :math:`\ D = {d\over dx}\ \,` viewed as a linear transformation of the space :math:`\,V\ ` of real polynomials of degree :math:`\,n\ ` into the space :math:`\,W\ ` of polynomials of degree :math:`\,n-1.\ ` The matrix associated with this operation in natural bases of spaces :math:`\ V\ \,` and :math:`\, W\ ` is given by :eq:`MBC_D`. If :math:`\ v\,=\,a_0\,+\,a_1\,x\,+\,a_2\,x^2\,+\,a_3\,x^3\,+\,\ldots\,+\,a_n\,x^n\,\in V,` then :math:`\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 :math:`\,v\ \,` and :math:`\, w:` .. math:: \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] is precisely the relation :eq:`fund` in Theorem 10. :math:`\;` 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 :math:`\,K\,`): * | :math:`n`-dimensional space :math:`\,V\,` with basis :math:`\ \mathcal{B}=(v_1,\,v_2,\,\dots,\,v_n)\,,\ ` | :math:`m`-dimensional space :math:`\,W\,` with basis :math:`\ \mathcal{C}=(w_1,\,w_2,\,\dots,\,w_m)\,;` * | space :math:`\ \text{Hom}(V,W)\ ` of linear transformations of the space :math:`\ V\ ` into the space :math:`\ W;` * | space :math:`\ M_{m\times n}(K)\ ` of rectangular matrices with the entries from the field :math:`\ K. \,` .. Istotę przyporządkowania przekształceniom z :math:`\,\text{Hom}(V,W)\,` macierzy z :math:`\,M_{m\times n}(K)\,` przedstawia :math:`\;` .. admonition:: Theorem 11. :math:`\\` The mapping .. math:: 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 :math:`\ \text{Hom}(V,W)\,` and :math:`\, M_{m\times n}(K).` :math:`\;` **Proof** is preceded with a reminder of definitions of operations on linear transformations which make :math:`\,\text{Hom}(V,W)\,` a vector space. If :math:`\,F_1,F_2,F\in\text{Hom}(V,W),\ a\in K,\,` then .. math:: :nowrap: \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 :math:`\,M_{\mathcal{B}\mathcal{C}}\ ` is an isomorphism, we have to prove its additivity, homogenity and bijectivity. a. Additivity. :math:`\,` Let :math:`\,F_1,F_2\,\in\,\text{Hom}(V,W).\ ` Then he :math:`\,j`-th column of the matrix :math:`\,M_{\mathcal{B}\mathcal{C}}(F_1+F_2)` .. math:: 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 :math:`\,j`-th columns of the matrices :math:`\ M_{\mathcal{B}\mathcal{C}}(F_1)\ ` and :math:`\ \,M_{\mathcal{B}\mathcal{C}}(F_2)\,,\ \ j=1,2,\dots,n.\ \,` Hence, .. math:: M_{\mathcal{B}\mathcal{C}}(F_1+F_2)\ =\ M_{\mathcal{B}\mathcal{C}}(F_1) \,+\,M_{\mathcal{B}\mathcal{C}}(F_2)\,. b. Homogenity. Let :math:`\,F\in\text{Hom}(V,W),\ \ a\in K.\ \,` Then the :math:`\,j`-th column of the matrix :math:`\,M_{\mathcal{B}\mathcal{C}}(aF)` .. math:: I_{\mathcal{C}}\,[\,(aF)(v_j)\,]\ =\ I_{\mathcal{C}}\,[\,a\cdot F(v_j)\,]\ =\ a\cdot I_{\mathcal{C}}\,[\,F(v_j)\,] is the :math:`\, j`-th column of the matrix :math:`\,M_{\mathcal{B}\mathcal{C}}(F)\,,\ \ j=1,2,\dots,n, \,` multiplied by :math:`\,a.\,` Hence, .. math:: M_{\mathcal{B}\mathcal{C}}(a\,F)\ =\ a\,M_{\mathcal{B}\mathcal{C}}(F)\,. c. Bijectivity. We have to show that every matrix :math:`\,\boldsymbol{F}\in M_{m\times n}(K)\,` is associated with exactly one mapping :math:`\,F\in\text{Hom}(V,W).\,` Indeed, columns of the matrix :math:`\boldsymbol{F}\,` determine (by the coordinates in the basis :math:`\, \mathcal{C}\,`) :math:`\,` images :math:`\, Fv_j\,` of basis vectors :math:`\,v_j\in\mathcal{B},\,` and thus (cf. Corollary to Theorem 5.) :math:`\,` the transformation :math:`\ F\ ` is uniquely defined. :math:`\;` On the basis of Theorem 8. we may now write .. admonition:: Corollary. If :math:`\,V\ \,` and :math:`\, W\ ` are finite dimensional vector spaces over a field :math:`\,K,\ \,` then .. math:: \dim\,\text{Hom}(V,W)\ =\ \dim\,V\,\cdot\,\dim\,W\,. :math:`\;` We consider one more case: :math:`\,V=K^n\,` with the canonical basis :math:`\,\mathcal{E}=(\boldsymbol{e}_1,\boldsymbol{e}_2,\dots,\boldsymbol{e}_n)\,,` :math:`\,W=K^m\,` with the canonical basis :math:`\, \mathcal{F}=(\boldsymbol{f}_1,\boldsymbol{f}_2,\dots,\boldsymbol{f}_m),\,` and :math:`\, F\in\text{Hom}(K^n,K^m).` A matrix of the transformation :math:`\,F\,` in the canonical bases :math:`\, \mathcal{E}\,` and :math:`\, \mathcal{F}\,` is of the form .. math:: 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 :math:`\,K^m\ ` each vector is a column of its coordinates in the canonical basis: :math:`\ \ I_{\mathcal{F}}(\boldsymbol{w})=\boldsymbol{w},\ \ \boldsymbol{w}\in K^m.\ ` If we denote the matrix of the transformation :math:`\,F\ ` in the canonical basis simply by :math:`\,M(F),\ ` we obtain a simplified formula: .. math:: M(F)\ =\ [\,F\boldsymbol{e}_1\,|\,F\boldsymbol{e}_2\,|\,\dots\,|\,F\boldsymbol{e}_n\,]\,, which has been introduced earlier in the equation :eq:`intro`. The formula :eq:`fund` in Theorem 10. takes now the form .. math:: \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\,.