3.3: Solution of the Homogeneous equation ax+by = 0
- Page ID
- 60310
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Proposition 3.5
The general solution of the homogeneous equation \(r_{1}x+r_{2}y = 0\) is given by
\[\begin{array} {ccc} {x = k \frac{r_{2}}{\gcd(r_{1}, r_{2})}}&{and}&{y = -k \frac{r_{2}}{\gcd(r_{1}, r_{2})}} \nonumber \end{array}\]
where \(k \in \mathbb{Z}\).
- Proof
-
On the one hand, by substitution the expressions for \(x\) and \(y\) into the homogeneous equation, one checks they are indeed solutions. On the other hand, \(x\) and \(y\) must satisfy
\[\frac{r_{1}}{\gcd(r_{1}, r_{2})} x = - \frac{r_{2}}{\gcd(r_{1}, r_{2})} y \nonumber\]
The integers \(r_{i}\) (for \(i\) in \(\{1,2\}\)) have greatest common divisor equal \(\gcd(r_{1},r_{2})\) to \(1\). Thus Euclid’s lemma applies and therefore \(\frac{r_{1}}{\gcd(r_{1}, r_{2})}\) is a divisor of \(y\), while \(\frac{r_{2}}{\gcd(r_{1}, r_{2})}\) is a divisor of \(x\).
A different proof of this lemma goes as follows. The set of all solution in \(\mathbb{R}^{2}\) of \(r_{1}x+r_{2}y = 0\) is given by the line \(l(\xi) = \begin{pmatrix} {r_{2}}\\ {-r_{1}} \end{pmatrix} \xi\). To obtain all its lattice points (i.e., points that are also in \(\mathbb{Z}^2\)), both \(r_{2} \xi\) and \(-r_{1} \xi\) must be integers. The smallest positive number \(\xi\) for which this is possible, is
\[\xi = \frac{1}{\gcd(r_{1}, r_{2})} \nonumber\]
Here is another homogeneous problem that we will run into. First we need a small update of Definition 1.2.
Definition 3.6
Let \(\{b_{i}\}^{n}_{i=1}\) be non-zero integers. Their greatest common divisor, \(\mbox{lcm} (b_{1}, \cdots, b_{n})\), is the maximum of the numbers that are divisors of every \(b_{i}\); their least common multiple, \(\gcd(b_{1}, \cdots, b_{n})\), is the least of the positive numbers that are multiples of of every \(b_{i}\).
Surprisingly, for this more general definition, the generalization of Corollary 2.15 is false. For an example, see exercise 2.6.
Corollary 3.7
Let \(\{b_{i}\}^{n}_{i=1}\) be non-zero integers and denote \(B = \mbox{lcm} (b_{1}, \cdots b_{n})\). The general solution of the homogeneous system of equations \(x = _{b_{i}} 0\) is given by
\[x =_{B} 0 \nonumber\]
- Proof
-
From the definition of \(\mbox{lcm} (b_{1}, \cdots, b_{n})\), every such \(x\) is a solution. On the other hand, if \(x \ne _{B} 0\), then there is an \(i\) such that \(x\) is not not a multiple of \(b_{i}\), and therefore such an \(x\) is not a solution.