Consider the following triangular system: where b is a real number. (x+y+z+w = 2 -6x+5w 3z-w 2w = b2 "I -4 a. For what value(s) of b, if any at all, does this system have a unique solution? Briefly justify your answer with a short explanation or short calculation. b. For what value(s) of b, if any at all, does this system have infinitely many solutions? Briefly justify your answer with a short explanation or short calculation. c. For what value(s) of b, if any at all, does this system have no solution? There's no need to justify your answer to (c) as it will be the complement of your answers to (a) and (b). Consider the following systems of square linear systems: x+3y-z =4 a. 4x-y+2z =8. (2x-7y+4z = -3 x+3y-z =4 b. 4x-y+22 = 8. (2x-7y+4z = 0 x+3y-z =4 C. 4x-y+2z = 8. (2x-7y+6z=0 Which of these systems has no solution? (There is only one.) What are the solutions of the remaining two systems? Exercise 1. (CS1.1) Component Skills Exercises A chemist must determine the coefficients x; (for 1 j 7) that balance the reacti x1PCH4 + x2KMnO4 + 3H2SO4 x4PCHO2 + 5K2SO4 + x6MnSO4 + x; -> In order to balance this reaction, there must be an equal number of each element the reaction. Use this information to derive a linear system satisfied by the unkn xj. What are the dimensions of this system? Remark: Please derive an equation for each element. Note that some of your be redundant. For sake of consistency, keep these redundant equations in your syst Exercise 2. (CS1.2) Consider the following triangular system: x + y + z = 2 a22y+2% = b . 3z = 9 a. Suppose a22 = 1 and b = 6. What is the solution of this system? b. Suppose a220 and b = 6. What is the solution of this system? c. Suppose a220 and b = 0. What is the solution of this system? Exercise 5. (CS1.3) Express the linear system in Exercise 1 in matrix-vector form and as an augmented matrix. Exercise 6. (CS1.4) From Olver and Shakiban, complete Exercises 1.2.7(b), (c), (f), and (h). For (h), I is the ap- propriately sized identity matrix. Remark: Unless very special circumstances are at play, remember AB + BA.