Application to Homogeneous Systems of Equations

We will now apply a theory of linear transformations of vector spaces to describe the set of solutions of homogeneous system of linear equations over a field \(\,K\):

(1)\[\begin{split}\begin{array}{r} a_{11}\,x_1\; + \ \,a_{12}\,x_2\; + \ \,\ldots\ + \ \;a_{1n}\,x_n \ \, = \ \ 0 \\ a_{21}\,x_1\; + \ \,a_{22}\,x_2\; + \ \,\ldots\ + \ \;a_{2n}\,x_n \ \, = \ \ 0 \\ \qquad\qquad\,\ldots\qquad\quad\ldots\qquad\quad\ldots\qquad\ldots\qquad\ \ \,\ldots \\ a_{m1}\,x_1\; + \ \,a_{m2}\,x_2\; + \ \,\ldots\ + \ \;a_{mn}\,x_n \ \, = \ \ 0 \end{array}\end{split}\]

The system has a matrix form \(\quad\boldsymbol{A}\boldsymbol{X}\,=\,\boldsymbol{0}\,,\quad\) where

\[\begin{split}\boldsymbol{A}\ =\ \left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{array} \right]\,,\quad \boldsymbol{X}\ =\ \left[\begin{array}{c} x_1 \\ x_2 \\ \dots \\ x_n \end{array}\right]\,,\quad \boldsymbol{0}\ =\ \left[\begin{array}{c} 0 \\ 0 \\ \dots \\ 0 \end{array}\right]\in K^m\,.\end{split}\]

We define the set of solutions of the system (1) as

\[S_0\ :\,=\ \{\,\boldsymbol{X}\in K^n:\ \boldsymbol{A}\boldsymbol{X}=\boldsymbol{0}\,\}\,.\]

Of course, \(\ \,S_0\subset K^n.\ \,\) The properties of the set \(\ S_0\ \) are better described in

Theorem 9. \(\\\)

The set of solutions of homogeneous system of linear equations (1) is a vector space over a field \(\,K\ \) (subspace of the space \(\,K^n\)), \(\,\) whose dimension equals the difference of the number of unknowns and the rank of the coefficient matrix \(\boldsymbol{A}:\)

(2)\[S_0\,<\,K^n,\qquad\dim\,S_0\,=\,n-\text{rk}\,\boldsymbol{A}\,.\]

Proof.

The subset \(\ S_0\ \) of the space \(\,K^n\ \) is a subspace because it is closed under addition of vectors and their multiplications by scalars from the field \(\,K.\ \) Indeed,

if \(\qquad\boldsymbol{X}_1,\,\boldsymbol{X}_2\,\in\,S_0: \qquad\boldsymbol{A}\boldsymbol{X}_1=\,\boldsymbol{0}\,, \quad\boldsymbol{A}\boldsymbol{X}_2=\,\boldsymbol{0}\,,\)

then \(\qquad \boldsymbol{A}\,(\boldsymbol{X}_1+\boldsymbol{X}_2)\ =\ \boldsymbol{A}\boldsymbol{X}_1+\boldsymbol{A}\boldsymbol{X}_2\ =\ \boldsymbol{0}\,, \qquad \boldsymbol{A}\,(c\,\boldsymbol{X}_1)\ =\ c\,(\boldsymbol{A}\boldsymbol{X}_1)\ =\ \boldsymbol{0}\,,\)

so \(\qquad \boldsymbol{X}_1+\boldsymbol{X}_2\,\in\,S_0\,,\qquad c\,\boldsymbol{X}_1\in S_0\,,\quad c\in K\,.\)

For the proof of the second part of the hypothesis, denote \(\ r\,:\,=\,\text{rk}\,\boldsymbol{A}\,.\ \) Of course, \(\ r\le m,n\,.\)

The matrix \(\boldsymbol{A}\ \) has \(\ r\ \) linearly independent rows and the same number of linearly independent columns. Without loss of generality, we may assume that the linearly independent set is determined by first \(\ r\ \) rows, \(\,\) and also by first \(\ r\ \) columns.

If \(\ m>r,\ \) then we discard last \(\ m-r\ \) equations because each of them is a linear combination of the first \(\ r\ \) equations.

As a starting point of the further discussion we may take a set of \(\ r\ \) equations with \(\ n\ \) unknowns, \(\,\) where \(\ r\le n.\ \) In this situation there are two possibilities.

I.) \(\,\) If \(\ r=n,\ \) we have a system with a square non-degenerate matix \(\boldsymbol{A}.\ \) This is a Cramer system which has only a zero solution: \(\ S_0=\{\boldsymbol{0}\}.\ \) In this case the equation (2) is fulfilled: \(\ 0=\dim\,S_0=n-r.\)

II.) \(\,\) Let \(\ r<n.\ \) We treat the unknowns indexed by numbers greater than \(\ r\ \) as parameters: \(\ x_k\rightarrow s_k,\ k=r+1,\dots,n,\ \,\) and then solve a Cramer system with the unknowns \(\ \,x_1,\,\dots,\,x_r:\)

(3)\[\begin{split}\begin{array}{c} a_{11}\,x_1\; + \ \,\ldots\ \, + \ \;a_{1r}\,x_r \ \, = \ \ -\ a_{1,r+1}\,s_{r+1}\; - \ \,\ldots\ \, -\ a_{1n}\,s_n \\ a_{21}\,x_1\; + \ \,\ldots\ \, + \ \;a_{2r}\,x_r \ \, = \ \ -\ a_{2,r+1}\,s_{r+1}\; - \ \,\ldots\ \, -\ a_{2n}\,s_n \\ \qquad \ldots\qquad\quad\ldots\qquad\quad\ldots\qquad\quad \ldots\qquad\quad\ldots\qquad\quad\ldots\qquad\quad \\ a_{r1}\,x_1\; + \ \,\ldots\ \, + \ \;a_{rr}\,x_r \ \, = \ \ -\ a_{r,r+1}\,s_{r+1}\; - \ \,\ldots\ \, -\ a_{rn}\,s_n \end{array}\end{split}\]

Let \(\ (c_1,\dots,c_r)\ \) be a tuple of the values of the unknowns \(\ x_1,\,\dots,\,x_r\,,\ \) corresponding to fixed values of the parameters \(\ s_{r+1},\,\dots,\,s_n.\\\) Then the tuple \(\ (c_1,\dots,c_r,\,s_{r+1},\dots,\,s_n)\ \) is one of solutions of the sytem fo equations (1).

Define a mapping \(\ \Phi:\ S_0\rightarrow K^{n-r}\ \) as follows:

\[\begin{split}\Phi:\qquad S_0\,\ni\, \left[ \begin{array}{c} c_1 \\ \dots \\ c_r \\ s_{r+1} \\ \dots \\ s_n \end{array} \right] \quad\rightarrow\quad \Phi \left[ \begin{array}{c} c_1 \\ \dots \\ c_r \\ s_{r+1} \\ \dots \\ s_n \end{array} \right] \ :\,=\ \left[ \begin{array}{c} s_{r+1} \\ \dots \\ s_n \end{array} \right] \,\in\,K^{n-r}\,.\end{split}\]

It is easy to check that \(\ \Phi\ \) is a linear mapping. Moreover, because each tuple of the values of the parameters \(\ s_{r+1},\,\dots,\,s_n\ \) corresponds to exactly one solution \(\ (c_1,\dots,c_r)\ \) of the Cramer system (3), \(\ \Phi\ \) is also a bijection.

Hence, the mapping \(\ \Phi,\ \) as a bijective homomorphism, is an isomorphism of the solution space \(\ S_0\ \) onto the space \(\ K^{n-r}.\ \) By Theorem 8. we obtain the hypothesis (2):

\[\dim\,S_0\,=\,\dim\,K^{n-r}\,=\,n-r\,.\]