(1) (1.1: Geometry) For each part below, give an example of a linear system of equations in two variables that has the given property. In each case, draw the lines corresponding to the solutions of the equations in the system. (a) has no solution (b) has exactly one solution (c) has infinitely many solutions (i) Either add or remove equations in (b) to make an inconsistent system. (ii) Either add or remove equations in (b) to create infinitely many solutions. (iii) Either add or remove equations in (b) so that the solution space remains unchanged. In each of (i) - (iii) justify your action in words. (2) 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 Aq + (1 - A)p as A 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 R" consists of all points of the form Aq + (1 - Ajp as A varies over all real numbers. (ii) Suppose p = (1,3, 4) and q = (5, 8, 12) are solutions to the linear system of equations: 0:2 = 03 = 04 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: and, + ajoke + ands = 1 1I = Check that any multiple of p, i.e., a vector of the form A(1, 3,4) where A 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. (3) (1.2: Solving linear equations) Consider the following linear system with a and b unknown non-zero constants. + 13 = 2:1 1 813 (a) For what values of a and b does the system have infinitely many solutions? (b) Given an example of a and b where the system has exactly one solution. (c) Give an example of a and b for which the system has no solutions. (4) (after 2.1) Find a 3 x 4 matrix A, in reduced echelon form, with free variable d'3, such that the general solution of the equation Ax = is X = + s where s is any real number. (5) (after 2.2) (a) The set P : 201 - 22 + 413 = 0 is a plane in R3. Find two vectors uj, u, ( R' so that span (un, u2} = P. Explain your answer. (b) Consider the three vectors us - 7 . .- - 1 -- [81 be an arbitrary vector in R3. Use Gaussian elimination to determine which vectors b are in span (U1, u2, us}- Without further calculation, conclude that span { uj, u2, us} is a plane in R3 and identify an equation of the plane in the form ac, + br2 + CT3 = 0