7. (a) A system of linear equations with fewer equations than unknowns is often underdetermined system. Suppose that such a system is consistent. Ext it must have an infinite number of solutions. (b) A system of linear equations with more equations than unknowns is off a overdetermined system. Can such a system be consistent? Give an exa system with three equations in two unknowns to illustrate your answer. 8. A system of linear equations is called homogeneous if the right hand side of e

Let's break down the problems step by step. ### Problem 7 #### (a) Underdetermined System An underdetermined system of linear equations has fewer equations than unknowns. If such a system is consistent, it means that there is at least one solution. However, because there are more unknowns than equations, there are not enough constraints to uniquely determine all the unknowns. Therefore, this system must have an infinite number of solutions. **Conclusion for (a):** Yes, if a consistent underdetermined system has fewer equations than unknowns, it must have an infinite number of solutions. #### (b) Overdetermined System An overdetermined system of linear equations has more equations than unknowns. Such a system can be consistent if all the equations do not contradict each other. However, it is also possible for an overdetermined system to be inconsistent if the additional equations introduce contradictions. **Example of a consistent overdetermined system:** Consider the following system of three equations in two unknowns \(x\) and \(y\): 1. \(x + y = 2\) 2. \(2x + 2y = 4\) 3. \(3x + 3y = 6\) In this case, all three equations represent the same line in the \(xy\)-plane. Therefore, they are consistent, and the system has infinitely many solutions along the line defined by \(x + y = 2\). **Conclusion for (b):** Yes, an overdetermined system can be consistent. The example provided illustrates this. ### Problem 8 A system of linear equations is called homogeneous if the right-hand side of each equation is equal to zero. In other words, a homogeneous system can be expressed in the form: \[ A\mathbf{x} = \mathbf{0} \] where \(A\) is a matrix of coefficients, \(\mathbf{x}\) is a vector of unknowns, and \(\mathbf{0}\) is the zero vector. Homogeneous systems always have at least one solution, which is the trivial solution where all unknowns are zero. If the system has more unknowns than equations, it will also have infinitely many solutions. If you have any further questions or need additional examples, feel free to ask!