1.1 Introduction to Linear Systems and Row Reduction

Transcription

1 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION. Introduction to Linear Systems and Row Reduction MATH 9 FALL 98 PRELIM # 9FA8PQ.tex.. Solve the following systems of linear equations. If there is no solution, show why. If there are innitely many solutions, give a general expression. x A b) c) x x x MATH 9 FALL 98 FINAL # 9FA8FQ.tex.. a) Find all possible solutions ~x of B~x = ~c, where B = b) For the system ~x = ~ b, where = A A and ~c = ; determine all vectors ~ b for which the system possesses nontrivial solutions ~x. MATH 9 SPRING 98 PRELIM # 9SP8PQ.tex.. onsider the system : x + y z + w = x + z + w = x + y + z + w = x + y + z + w = a) Find all the solutions to this system. b) Find a basis for the vector space of solutions to the system above. You need not prove this is a basis. c) What is the dimension of the vector space of solutions above? d) Is the the vector x y z w = a solution to the above system?

2 MATH 9 SPRING 98 FINAL # 9SP8FQ.tex.. Find the general solution, or else show that the system has no solutions: x x + x = x + x x + x = x x = MATH 9 FALL 98 FINAL # 9FA8FQ.tex.. Find the general solution of the system x + x + x = x + x + x + x = x + x x + x = 8 and express your answer in vector form. MATH 9 FALL 98 FINAL # 9FA8FQ.tex.. a) Solve the linear system A~x = ~ b, where and ~ b = b) Solve the linear system A~x = ~, where Express your answer in vector form, and give a basis for the space of solutions. MATH 9 SPRING 98 PRELIM # 9SP8PQ.tex..* Find all solutions to: x + z = y + z = y 8z = 9 9 :

3 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 FALL 98 PRELIM # 9FA8PQ.tex..8 a) Determine the row-reduced form of the matrix: 8 9 : b) Find the general solution of A~u = ~, where ~u = u u u u u and ~ = MATH 9 FALL 98 MAKE-UP # 9FA8MUQ.tex..9 Use row reduction to either nd the solution or show that no solution exists for the system x x = x + x + x = x x + x = MATH 9 SPRING 989 PRELIM # 9SP89PQ.tex.. onsider the system of equations, : x + x + x = x + x x = x + x x = a) Find all solutions, if any exist, of the system. b) Is the set of vectors given by, ; linearly independent or dependent? ; and MATH 9 SUMMER 989 PRELIM # 9SU89PQ.tex.. a) Find all solutions to x + x x + x = x + x x + x = x + x x + x = using only the row reduction method.

4 MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex..* Find all the solutions. Write your answers in vector form. a) x x x = x x x = x + x + x = b) x + x + x = x x x = x + x + x = c) x + x x = x x x = x x + x = MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. onsider A~x = ~ b. Where Ais a) Solve for ~x given ~ b = A. b) Find a basis for the null space of A. c) Without carrying out explicit calculation, does a solution exist for any ~ b in V? A : MATH 9 FALL 99 PRELIM # 9FA9PQ.tex.. Solve for the x matrix X if X = : MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Here a) Find A. b) Find Xif AX = c) Find ~vif A~v =...

5 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 FALL 99 PRELIM # 9FA9PQ.tex.. Find all solutions of the system A~x = ~ b if a) ~ b =. b) ~ b = Express all answers in vector form. 9 MATH 9 FALL 99 FINAL # 9FA9FQ.tex.. a) Find the eigenvalues and eigenvectors of the matrix B = : and. Find a non singular matrix such that A = D where b) Let D is a diagonal matrix. Find and D. c) For which value of a does the system of equations x + y + z = a x + y + z = x + z = has at least one solution? Explain your answer. MATH 9 SPRING 99 FINAL # 9SP9FQ.tex..8 a) Find the general solution, and write your answer as a particular solution plus the general solution of the associated homogeneous system. x y + z = x y + z = x + y + z = b) heck your answer for part a. c) Find the inverse of the d) heck your answer for part c. A :

6 MATH 9 SPRING 99 FINAL # 9SP9FQ.tex..9 onsider the matrix a) Find the vectors b b b A : A such that a solution x of the equation A~x = ~ b exists. b) Find a basis for the column space <(A) of A. c) It is claimed that <(A) is a plane in <. If you agree, nd a vector ~n in < that is normal to this plane. heck you answer. d) Show that ~n is perpendicular to each of the columns of A. Explain carefully why this is true. MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. (True/false) The following properties hold for the matrix a) If AM = AN then M = N, where M and N are x matrices. b) A has an inverse. c) A is in reduced row echelon form. d) A is equal to the matrix B = e) A and B are row equivalent. f) A and B have the same row reduced form. g) (A T ) T = A. h) B T BA T. :.

7 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex..* If the reduced row echelon form A then the general solution of the system is = = x y + z w = x + y + z w = x y z w = A d) A, b) = A + t = = 9=8 = = =8 = = A +t A, e) A, c) A + MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex..* The reduced row echelon for of A, A, c) A @ A, A : (Note! There is one and only one correct answer.) MATH 9 FALL 99 PRATIE # 9FA9PPQ.tex.. Find the general solution in vector form for the equations x + y + z w = x + y + z + w = x y + z w = A : ; A A = = = A, A,

8 8 MATH 9 FALL 99 PRELIM # 9FA9PQ.tex.. Use Gauss-Jordan elimination to nd all solutions of in the cases that b b x + y + z = b x + y + z = b x + y + 9z = b A and b b MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Find the general solution of the wystem of equations x + x = x + x + x = x + x + x + x = x + x x + x = A MATH 9 SPRING 99 FINAL # 9SP9FQ.tex.. Find the general solution of the system of equations x + x + x = x + x + x + x = x + x x + x = 8 MATH 9 FALL 99 PRELIM # 9FA9PQ.tex..* a) Find the general solution of the wystem of equations x + x = x + x + x = x + x + x + x = x + x x + x = b) Verify your solution. MATH 9 FALL 99 FINAL # 9FA9FQ.tex..8* a) Find the general solution, in vector form, of the equation A~x = ~ b where Verify your solution. and ~ b = :

9 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION 9 MATH 9 SPRING 99 FINAL # 9SP9FQ.tex..9* onsider the system x + x + x x = x + x + x + x = x + x + x = A solution of these equations is: a) The trivial soution. b) x = 9; x = ; x = ; x = c) x = ; x = ; x = ; x = d) The system has no solution. e) None of the above. MATH 9 SPRING 99 FINAL # 9SP9FQ.tex..* onsider the system x + z = x + y + z = x y + (a a)z = a 8 The value of a for which the system has innitely many solution is: a) b) c) none d) e) none of the above. MATH 9 FALL 99 PRELIM # 9FA9PQ.tex..* Matrix algebra. Let [A] be the matrix and let ~ b = and let ~c = a) Find all solutions ~x to A~x = ~ b and check your answer by substitution. b) Find all solutions ~x to A~x = ~c and check your answer by substitution. c) Give a reason why you believe that A does or does not exist. MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Find the general solution of the linear system x = + x + x + x + x = x + x =

10 9, ~ b =, and ~x = x x MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex..* Let x a) Find the characteristic polynomial det(a I) of A, and nd all eigenvalues. (hint : 9 is one factor of the polynomial.) b) nd an eigenvector for each eigenvalue. c) Write the augmented matrix for the system of equations A~x = ~ b and solve the system by row operations. MATH 9 SPRING 99 FINAL # 9SP9FQ.tex.. Solve the following system for x ; x ; x ; x and express the general solution in parametric form. x + x x = + x + x = x x = MATH 9 FALL 99 PRELIM # 9FA9PQ.tex.. a) onsider the problem A~x = ~ b, A ; and ~ b A : Determine the general solution to this problem, in vector form. b) Find a by matrix B, which is not the zero matrix, with B =. MATH 9 FALL 99 FINAL # 9FA9FQ.tex.. Find the general solution of these equations in vector parametric form. x x = x + x = x x =.

11 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 SPRING 998 PRELIM # 9SP98PQ.tex.. a) Write the solution set of the system in parametric form. b) With x x x = x x = x + x + x = A nd all solutions to the system A! x = : ; c) True or False? i) The columns of A are linearly independent. ii) The solution set of A! x =! b is all vectors of the form! w =! p + v! h where v! h is any solution of A v! h =! and A! p =! b. MATH 9 SPRING 998 PRELIM # 9SP98PQ.tex..8* Find all solution to the following matrix equation A~x = ~ b where for each of the following values of ~ b: a) ~ b = b) ~ b = c) ~ b =

12 MATH 9 FALL 998 PRELIM # 9FA98PQ.tex..9 a) Write the following system of equations as (i) a vector equation (ii) as a matrix equation. x + x = x x + 9x = x + x x + x = b) Find all solutions to the linear system x + x x = x + x + x = x + x + x = c) Does the above (b) have a solution for any right hand side? d) Let ~u =, ~u =, ~ b = h For what value(s) of h is ~ b in the plane spanned by f~u ; ~u g? MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Let ; B = a) Find all ~x for which ~x = ~ b, where ~ b = : ; = :

13 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 FALL 998 PRELIM # 9FA98PQ.tex..* The reduced echelon form of the matrix B =. a) What is the rank of A. b) What is the dimenstion of the column space of A? c) What is the dimension of the null space of A? d) Find a solution to A~x =. e) Find the general solution to A~x =. is f) What is the row space of A? g) Would any of your answers above change if you changed A by randomly changing of its entries in the nd, third, and fourth columns to dierent small integers and the corresponding reduced echelon form for B was presented? (yes?, no?, probaly? probaly not?,?) MATH 9 FALL 998 FINAL # 9FA98FQ.tex.. onsider A~x = ~ b with 8. The augmented matrix of this system is. 8 and b = which is row equivalent to a) What are the rank of A and dim nul A? (Justify your answers.) b) Find bases for col A, row A, and nul A. c) What is the general solution ~x to A~x = ~ b with the given A and ~ b? d) Select another ~ b for which the above system has a solution. Give the general solution for that ~ b.

14 MATH 9 SPRING? FINAL # 9SPUFQ.tex..* Find the general soliution of the following linear system and express it in vector form x y + z w = 8 x + y z + w = x + y + z w = x y + z w = MATH 9??? FINAL # 9UFQ.tex..* a) Give all solutions of the following system in vector form. x + x = x x + x = x + x = b) What is the null space of the matrix of coecients of the unknowns in a)? MATH 9 SPRING 98 PRELIM # 9SP8PQ.tex.. a) Write the system of equations x + x + x = x + x + x = x + x + x = in the form A~x = ~ b. b) Find the det A for A in part (a) above. c) Does A exist? d) Solve the above system of equations for ~x = e) Let B x x x A.. Find A B (i.e. calculate the product AB). MATH 9 FALL 99 FINAL # 9FA9FQ.tex.. Write the general solution in vector form: x y + z + w = x + y z + w = x + y z w = x + y z + w = note: This problem is the same as MATH 9 SUMMER 99 PRELIM / #

15 .. INTRODUTION TO LINEAR SYSTEMS AND ROW REDUTION MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Find the general solution solution in vector form for the equations x + y + z w = x + y + z + w = x y + z w = MATH 9 SUMMER 99 PRELIM / # 9SU9PQ.tex..8 Find the general solution of the equations x y + z + w = x + y z + w = x + y z w = x + y z + w = MATH 9 SUMMER 99 FINAL # 9SU9FQ.tex..9* a) Find the general solution, in vector form, of the equations A~x = ~ b b) Solve AX = B A ; ~ b A A and B MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. a) Solve the linear system: w x + y z = w + x y z = w + x y z = w x y z = Write the general solution in vector form as the sum of a particular solution plus the general solution of the associated homogeneous equation. b) heck your answer, and explain what you do to check. MATH 9 SPRING 99 PRELIM # 9SP9PQ.tex.. Two chemicals A and B, are reacting with each other. After one second has elapsed, 9% of chemical A stays chemical A, while % turns into chemical B; also, 8% of chemical B stays chemical B, while % turns into chemical A. Suppose that the system is in equilibrium, i.e. that there is no change in the amount in grams of chemical A or B from one second to the next. If there are grams of chemical A at equilibrium, how many grams of chemical B must there be? A

16 MATH 9 FALL 99 PRELIM # 9FA9PQ.tex.. A spaceship operator operates daily spaceship service between three planets, A; B; and. The matrix below shows the trac during a Monday. The numbers are fractions of the total number of spaceships that start at one location, and go to another destination. For example, the. means that % of the spaceships that start at travel to A that day. from A : : : : A A B The distribution of spaceships at planets A; B; and on Monday is,b; c. Find b and c such that the same distribution of space ships reappears the next day, on Tuesday. MATH 9 FALL 998 PRELIM # 9FA98PQ.tex.. onsider the chemical reaction (unbalanced as written below) H O + O! O + H O: Let x ; x ; x ; and x be the number of molecules of each compound (in the order give above). Find integers x ; x :x ; x that balance this reaction. Hint: If you order your elements and hence equations Oxygen arbon Hydrogen A ; O H A ; you will minimize th number of row operations. MATH 9 FALL 998 FINAL # 9FA98FQ.tex.. The kingdom of Ferrgrad has three primary industrial sectors: iron, railroad, and coal. Suppose that: To produce $ of steel, the steel sector consumes $. of steel, $. of railroad, and $. of coal. To produce $ of railroad transportation that rail sector consumes $. of steel, $. of rail, and $. of coal. To produce $ of coal, the coal sector consumes $. of steel, $. of rail, and $. of coal. Ferrograd does not use all its production in the various sectors to maintain the others. Additionally it exports S = $:x of steel, R = $:8x of railroad transportational services, = $:x of coal. Dene S; R; and to be the $ values of annual production of steel, rail, and coal. Set and do not solve a matrix equation that will tell you S; R; and, the values of the annual productions of the three sectors. [Hint: rst write three simultaneous equations, one for each output, which relate production to output.] to