Notice that if \(n=m\) or \(n m\). Hence if we are given a matrix equation to solve, and we have already solved the homogeneous case, then we need only find a single particular solution to the equation in order to determine the whole set of solutions. We call this the trivial solution. Specifically, \[\begin{array}{c} x = 0 \\ y = 0 + t \\ z = 0 + t \end{array}\] can be written as \[\left[ \begin{array}{r} x\\ y\\ z \end{array} \right] = \left[ \begin{array}{r} 0\\ 0\\ 0 \end{array} \right] + t \left[ \begin{array}{r} 0\\ 1\\ 1 \end{array} \right]\] Notice that we have constructed a column from the constants in the solution (all equal to \(0\)), as well as a column corresponding to the coefficients on \(t\) in each equation. There are less pivot positions (and hence less leading entries) than columns, meaning that not every column is a pivot column. Then, the solution to the corresponding system has \(n-r\) parameters. On the basis of our work so far, we can formulate a few general results about square systems of linear equations. Therefore, our solution has the form \[\begin{array}{c} x = 0 \\ y = z = t \\ z = t \end{array}\] Hence this system has infinitely many solutions, with one parameter \(t\). Denition 1 A linear system of equations Ax = b is called homogeneous if b = 0, and non-homogeneous if b 6= 0. It turns out that looking for the existence of non-trivial solutions to matrix equations is closely related to whether or not the matrix is invertible. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. At this point you might be asking "Why all the fuss over homogeneous systems?". 37 We know that this is the case becuase if p=x is a particular solution to Mx=b, then p+h is also a solution where h is a homogeneous solution, and hence p+0 = p is the only solution. Then, the number \(r\) of leading entries of \(A\) does not depend on the you choose, and is called the rank of \(A\). Matrices 3. Linear Algebra/Homogeneous Systems. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Consider the following homogeneous system of equations. Examine the following homogeneous system of linear equations for non-trivial solution. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. For instance, looking again at this system: we see that if x = 0, y = 0, and z = 0, then all three equations are true. \[\begin{array}{c} x + 4y + 3z = 0 \\ 3x + 12y + 9z = 0 \end{array}\] Find the basic solutions to this system. 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