The following exercises reveal structural properties of the set of solutions to a system of linear equations. the problems are set in r3, but the results extend to any r" (a) (i) suppose p = (1,3,4) and q = (5,8, 12) are two points in r3. show that the line joining p and q consists of all points of the form lq+ (1 - p as varies over all real numbers. (hint: think of the line as anchored p and going in directions (q-p) and -(q-p.) general statement: the line joining two points p and q in rn consists of all points of the form lq + (1 - \p as varies over all real numbers. (ii) suppose p = (1,3,4) and q=(5,8, 12) are solutions to the linear system of equations: 0 + 212.22 + 213.43 = 21 02111 + 022.2 2 + 23.13 = 2 03121 + 232.22 + 233.3 = 3 04141 +242.12 +243.13 = 24 check that all points on the line joining p and q are also solutions to the above system of equations. general statement: if a system of linear equations in n variables has two solutions, then all points on the line joining the two solutions are also solutions to the system. therefore, if a system of linear equations has at least two solutions, it has infinitely many solutions. (b) suppose p = (1, 3, 4) is a solution to the system of homogeneous equations: 0 + 212.22 + 213.23 = 0 021.11 + 0 +223.23 = 0 03121 + 232.22 + 233.13 = 0 04141 +242.22 +243.23 = 0 check that any multiple of p, i. e., a vector of the form (1,3,4) where is any real number, is also a solution of the system. is this an application of the previous question? general statement: if a homogeneous system of equations has a non-zero solution then it has infinitely many solutions