Orthogonality of Vectors ------------------------ We assume that :math:`\,V\ ` is a unitary or Euclidean vector space. .. Niech :math:`\,x,\,y\in V. ` If an inner product of vectors :math:`\,x,y\in V\ ` equals zero: :math:`\,\langle x,y\rangle=0\,,\ ` then we say that these vectors are :math:`\,` *orthogonal*. :math:`\,` Orthogonality is thus the generalization of the notion of perpendicularity of geometric vectors. Orthogonal Set of Vectors ~~~~~~~~~~~~~~~~~~~~~~~~~ .. admonition:: Definition. A set :math:`\ (x_1,x_2,\dots,x_r)\ ` of pairwise orthogonal non-zero vectors from the space :math:`\, V`, i.e. .. math:: x_i\neq \theta \qquad\text{and}\qquad \langle\,x_i,x_j\rangle=0\quad\text{for}\quad i\neq j\,,\qquad i,j=1,2,\dots,r\,, is called an :math:`\,` *orthogonal set*. :math:`\,` An orthogonal set of unit vectors (that is, of the vectors having norm :math:`\,1`) :math:`\,` is an :math:`\,` *orthonormal set*. Hence, an inner product of any two vectors from an orthonormal set :math:`\ (x_1,x_2,\dots,x_r)\ ` is given by .. math:: \langle\,x_i,x_j\rangle=\delta_{ij}\,,\quad\text{where}\quad\delta_{ij}\ \,=\ \, \left\{\ \begin{array}{cc} 1 & \text{for}\ \ i=j, \\ 0 & \text{for}\ \ i\ne j; \end{array} \right.\quad i,j=1,2,\ldots,r\quad \text{(the Kronecker delta).} A relation between othogonality and linear independence of vectors presents .. admonition:: Theorem 3. Every orthogonal set of vectors of the space :math:`\,V\ ` is linearly independent. **Proof.** :math:`\,` Assume that the set :math:`\ (x_1,x_2,\dots,x_r)\ ` of vectors from the space :math:`\,V\ ` is orthogonal: .. math:: :label: assumpt \langle\,x_i,x_i\rangle>0\,,\qquad\quad \langle\,x_i,x_j\rangle=0\quad\text{for}\quad i\neq j\,,\qquad\quad i,j=1,2,\dots,r\,. .. Dla wykazania liniowej niezależności tego układu przypuśćmy, że .. math:: \alpha_1\,x_1\,+\;\alpha_2\,x_2\,+\,\dots\,+\,\alpha_r\,x_r\ =\ \theta\,. Let :math:`\quad\alpha_1\,x_1\,+\;\alpha_2\,x_2\,+\,\dots\,+\,\alpha_r\,x_r\ =\ \theta\,.` After applying an inner product to both sides of the above equality with the vectors :math:`\ x_1,\;x_2,\,\dots,\,x_r\ \,` on the left hand side and :math:`\,` using linearity of an inner product with respect to the second variable, :math:`\,` we obtain .. .. math:: \alpha_1\,\langle x_1,x_1\rangle\ +\ \alpha_2\,\langle x_1,x_2\rangle\ +\ \ldots\ +\ \alpha_r\,\langle x_1,x_r\rangle\ =\ 0 \alpha_1\,\langle x_2,x_1\rangle\ +\ \alpha_2\,\langle x_2,x_2\rangle\ +\ \ldots\ +\ \alpha_r\,\langle x_2,x_r\rangle\ =\ 0 \dots\qquad\dots\qquad\dots\qquad\dots \alpha_1\,\langle x_r,x_1\rangle\ +\ \alpha_2\,\langle x_r,x_2\rangle\ +\ \ldots\ +\ \alpha_r\,\langle x_r,x_r\rangle\ =\ 0 .. math:: :nowrap: \begin{alignat*}{5} \alpha_1\,\langle x_1,x_1\rangle & {\,} + {\ } & \alpha_2\,\langle x_1,x_2\rangle & {\,} + {\ } & \ldots & {\ \ } + {\ } & \alpha_r\,\langle x_1,x_r\rangle & {\ } = {\ \,} & 0 \\ \alpha_1\,\langle x_2,x_1\rangle & {\,} + {\ } & \alpha_2\,\langle x_2,x_2\rangle & {\,} + {\ } & \ldots & {\ \ } + {\ } & \alpha_r\,\langle x_2,x_r\rangle & {\ } = {\ \,} & 0 \\ \dots\quad\ \ & & \dots\quad\ \ & & \ \ldots & & \dots\quad\ \ & & \\ \alpha_1\,\langle x_r,x_1\rangle & {\,} + {\ } & \alpha_2\,\langle x_r,x_2\rangle & {\,} + {\ } & \ldots & {\ \ } + {\ } & \alpha_r\,\langle x_r,x_r\rangle & {\ } = {\ \,} & 0 \end{alignat*} Conditions :eq:`assumpt` imply that :math:`\quad\alpha_1\,=\;\alpha_2\,=\;\dots\;=\,\alpha_r\ =\ 0\,.` Hence, the implication .. math:: \alpha_1\,x_1+\,\alpha_2\,x_2+\ldots+\,\alpha_r\,x_r\ =\ \theta \qquad\Rightarrow\qquad \alpha_1=\,\alpha_2=\ldots=\,\alpha_r\,=\,0\, is true, and thus the vectors :math:`\, x_1,\,x_2,\,\dots,\,x_r\,` are linearly independent. .. admonition:: Corollary. In :math:`\,n`-dimensional unitary or Euclidean vector space: 1. every orthogonal set of :math:`\,n\ ` vectors comprises a basis; 2. an orthogonal set of vectors cannot contain more than :math:`\,n\ ` vectors. Orthonormal Basis ~~~~~~~~~~~~~~~~~ .. admonition:: Definition. A basis of finite dimensional space :math:`\,V\ ` whose vectors comprise an orthogonal (orthonormal) set is called an *orthogonal basis* (resp. an *orthonormal basis*). .. **Zależności w bazie ortonormalnej.** Assume that a basis :math:`\,\mathcal{B}=(u_1,u_2,\dots,u_n)\ ` of the space :math:`\,V\ ` is orthonormal: .. math:: \langle\,u_i,u_j\rangle\,=\,\delta_{ij}\,,\qquad i,j=1,2,\dots,n. 1. Let :math:`\ \,v\,=\,\displaystyle\sum_{k\,=\,1}^n\ \alpha_k\,u_k\,.\ \,` Then, by definition of an inner product: .. math:: :label: ortho_1 \begin{array}{l} \displaystyle \langle\,u_i,v\,\rangle\ \,=\ \, \left\langle u_i\,,\ \sum_{k\,=\,1}^n\ \alpha_k\,u_k\right\rangle \ =\ \sum_{k\,=\,1}^n\ \alpha_k\,\langle u_i,u_k\rangle \ =\ \sum_{k\,=\,1}^n\ \alpha_k\,\delta_{ik}\ =\ \alpha_i\,; \\ \\ \blacktriangleright\quad\alpha_i\ =\ \langle\,u_i,v\,\rangle\,,\qquad i=1,2,\dots,n. \end{array} The :math:`\,i`-th coordinate of the vector :math:`\,v\ ` in basis :math:`\ \mathcal{B}\ ` is equal to an inner product of the :math:`\,i`-th vector from the basis :math:`\,\mathcal{B}\ ` and the vector :math:`\,v\,,\quad i=1,2,\dots,n.` 2. Let :math:`\quad v\,=\,\displaystyle\sum_{i\,=\,1}^n\ \alpha_i\,u_i\,,\ \ w\,=\,\displaystyle\sum_{j\,=\,1}^n\ \beta_j\,u_j\,:\quad I_{\mathcal{B}}(v)= \left[\begin{array}{c} \alpha_1 \\ \alpha_2 \\ \dots \\ \alpha_n \end{array}\right]\,,\ \ I_{\mathcal{B}}(w)= \left[\begin{array}{c} \beta_1 \\ \beta_2 \\ \dots \\ \beta_n \end{array}\right]\,.` .. math:: \begin{array}{rcl} \langle\,v,w\,\rangle & = & \left\langle\ \displaystyle\sum_{i\,=\,1}^n\ \alpha_i\,u_i\,, \ \displaystyle\sum_{j\,=\,1}^n\ \beta_j\,u_j\right\rangle\ \ =\ \ \displaystyle\sum_{i,j\,=\,1}^n\ \alpha_i^*\,\beta_j\ \langle\,u_i,u_j\rangle\ \ =\ \ \\ \\ & = & \displaystyle\sum_{i,j\,=\,1}^n\ \alpha_i^*\ \beta_j\ \delta_{ij}\ \ =\ \ \displaystyle\sum_{i\,=\,1}^n\ \alpha_i^*\,\beta_i\ \ =\ \ \langle\,I_{\mathcal{B}}(v),\,I_{\mathcal{B}}(w)\,\rangle\,; \\ \\ \blacktriangleright\quad\langle\,v,w\,\rangle & = & \langle\,I_{\mathcal{B}}(v), \,I_{\mathcal{B}}(w)\,\rangle\,. \end{array} An inner product of th vectors :math:`\,v\,` and :math:`\,w\,` (in a unitary or Euclidean space :math:`\,V`) :math:`\,` is equal to an inner product :math:`\,` (in the space :math:`\,C^n` or :math:`\,R^n,\,` respectively) :math:`\,` of column vectors representing coordinates of the vectors :math:`\,v\,` and :math:`\,w\,` in the basis :math:`\,\mathcal{B}.` 3. Let :math:`\,F\in\text{End}(V)\,,\ \ M_{\mathcal{B}}(F)=[\,\varphi_{ij}\,]_{n\times n}\,.\ ` By definition of matrix of a linear operator: .. math:: :label: ortho_3 \begin{array}{rcl} \langle\,u_i,Fu_j\rangle & = & \left\langle u_i\,,\,\displaystyle\sum_{k\,=\,1}^n\ \varphi_{kj}\,u_k\right\rangle\ \ = \ \ \displaystyle\sum_{k\,=\,1}^n\ \varphi_{kj}\,\langle u_i,u_k\rangle\ \ = \\ \\ & = & \displaystyle\sum_{k\,=\,1}^n\ \varphi_{kj}\ \delta_{ik}\ \ =\ \ \displaystyle\sum_{k\,=\,1}^n\ \delta_{ik}\ \varphi_{kj}\ \ = \ \ \varphi_{ij}\ ; \\ \\ \blacktriangleright\quad\varphi_{ij} & = & \langle\,u_i,Fu_j\rangle\,,\qquad i,j=1,2,\dots,n. \end{array} An element :math:`\,\varphi_{ij}\ ` of matrix of a linear operator :math:`\,F\,` in basis :math:`\,\mathcal{B}\ ` is equal to an inner product of the :math:`\,i`-th vector from the basis :math:`\,\mathcal{B}\ ` and the image :math:`\,` (under the transformation :math:`F`) of the :math:`\ \,j`-th vector from this basis, :math:`\ \ i,j=1,2,\dots,n.` It is worth to notice that while in an arbitrary basis :math:`\,\mathcal{B}=(v_1,v_2,\dots,v_n)\ ` coordinates :math:`\,\alpha_i\ ` of a vector :math:`\,v\ ` and elements :math:`\,\varphi_{ij}\ ` of matrix of a linear operator :math:`\,F\ ` are defined implicitely by relations .. math:: v\,=\,\sum_{i\,=\,1}^n\ \alpha_i\,v_i\,,\qquad Fv_j\,=\,\sum_{i\,=\,1}^n\ \varphi_{ij}\,v_i\,, \quad j=1,2,\dots,n\,, then in an orthonormal basis these quantities are given *explicitely* by formulae :eq:`ortho_1` :math:`\,` and :math:`\,` :eq:`ortho_3`. Moreover, the equation :eq:`ortho_1` implies that every vector :math:`\,v\in V\ ` may be written as .. math:: :label: coord v\ \,=\ \,\sum_{i\;\,=\ \,1}^n\ \alpha_i\,u_i\ =\ \sum_{i\,=\,1}^n\ \langle u_i,v\rangle\;u_i\,. .. admonition:: Definition. Let :math:`\,u,v\in V\,.\ ` If a vector :math:`\,u\ ` has norm :math:`\,1:\ \ \|u\|=1\,,\ \\` then an inner product :math:`\,\langle u,v\rangle\ ` is called a *coordinate of vector* :math:`\,v\ ` *on the axis* :math:`\,` u. The formula :eq:`coord` states that coordinates of vector :math:`\,v\ ` in an orthonormal basis :math:`\,\mathcal{B}=(u_1,u_2,\dots,u_n)\ ` are its coordinates on axes :math:`\,\text{u}_1,\,\text{u}_2,\,\dots,\,\text{u}_n\,.` **Example.** 1. An orthonormal basis of real 3-dimensional space of geometric vectors consists of three mutually perpendicular unit vectors :math:`\,\mathcal{E}=(\vec{e}_1,\vec{e}_2,\vec{e}_3).\ ` An inner product of vectors :math:`\,\vec{a}=\alpha_1\,\vec{e}_1+\alpha_2\,\vec{e}_2+\alpha_3\,\vec{e}_3\,,\ \vec{b}=\beta_1\,\vec{e}_1+\beta_2\,\vec{e}_2+\beta_3\,\vec{e}_3\ ` equals .. math:: \vec{a}\cdot\vec{b}\ =\ (\alpha_1\,\vec{e}_1+\alpha_2\,\vec{e}_2+\alpha_3\,\vec{e}_3)\cdot (\beta_1\,\vec{e}_1+\beta_2\,\vec{e}_2+\beta_3\,\vec{e}_3)\ =\ \alpha_1\,\beta_1\,+\,\alpha_2\,\beta_2\,+\,\alpha_3\,\beta_3\,. 2. An example of an orthonormal basis of a unitary space :math:`\,C^n\ ` (and also Euclidean space :math:`\,R^n`) is a canonical basis :math:`\,\mathcal{E}=(e_1,e_2,\dots,e_n),\ ` where the :math:`\,i`-th vector equals .. math:: e_i\ =\ \left[\begin{array}{c} 0 \\ \dots \\ 1 \\ \dots \\ 0 \end{array}\right] \begin{array}{c} \; \\ \; \\ \leftarrow\ i \\ \; \\ \; \end{array}\,, \qquad i=1,2,\dots,n\,.