Question
Please help with these. There are some that reference the text book, and I will deal with those. Thanks.
Note the contrapositive of the definition of one-to-one function given on page 141 of the text is: If a b then f(a) f(b), As wc know, the contrapositive is equivalent to (another way of saying) the definition of one-to-one. Consider the following function f: R R defined by f(x) = x2 - 9. Use the contrapositive of the definition of one-to-one function to determine (no proof necessary) whether f is a one-to-one function. Explain Compute f * f. Let g be the function g: R R defined by g(x) = x3 + 3. Find g-1 Use the definition of g -1 to explain why your solution, g -1 is really the inverse of g. Text page 169 number 34. part a. Let A = , B = and C = Compute: AC + BC (It is much faster if you use the distributive law for matrices first.) 2A-3A Perform the given operation for the following zero-one matrices. See the text, page 182. for the definition of the symbol O. Let A = Use the formula for finding the inverse of a 2x2 matrix given in exercise 19 page 184 to find A -1 . Use the definition of the inverse of a matrix given in the text just prior to exercise 18, page 184. to verify your answer in part (a). Read your notes carefully before attempting problems 5 and 6. Simply follow the examples). You must show all work. Here are a couple of online sites if you need them: Klan Academy Videos (click on Linear Algebra) and google Gaussian Elimination. Solve the following systems of equations using the method outlined in week 7 (Part I) of the notes. Your procedure should be in matrix form as was done in Example 3. ( Week 7 (Part 11. Systems Which Do Not Have a Unique Solution) of the notes. Page 5 number 2. Look up the definition (not the formula) for the inverse of a matrix. Note, the definition is given in your notes and in the text on page 184. prior to exercise 18. Use the definition of the inverse of a matrix to do exercise 18 on page 184 of the text. We know that matrix algebra behaves similar to (but not exactly the same as) regular algebra. The statements in parts a and b illustrate a couple of the differences between the two structures. Let A and B be arbitrary n x n matrices whose entries are real numbers. Use basic matrix laws only to expand to determine whether (A - B)(A + B) = A2 - B2. Explain all steps. Note the basic matrix laws are given in your notes, week 6 pg.4. Text page 153 number 12 parts c and d. Explain, but no proof necessary. This can be used for extra credit for either Exam 2 or exam 3. Everyone should try it. It's a nice application. Hand it in lor exam 2 when exam 2 is due or for exam 3 (when exam 3 is due) Project: Make up a meaningful network example. In the notes, week 7 pan III. on Network Analysis I have given you several different "flow models". Your project docs not have to mimic one of these. Try to make it different than those presented in the notes. Your example must be well defined. Define the problem clearly. Your example should involve solving a system of equations with an "infinite number of solutions". Actually, once you place restrictions on the variables (They may be integers and sec part f. ) the number of solution may be large but not an infinite number. Solve your system of equations any way you wish, but describe your solutions clearly. Explain your results with two or three examples. Hi at is. compute the flow when the free variable arc specific numbers. Interpret your results. Do they make sense? Do negative values make sense? Should there be a maximum capacity for each edge? If so. make up some reasonable maximum capacity for each edge.
Explanation / Answer
We don't have our note, so I'll let you do what's missing
1)
(a) 1!=-1 but f(1)=f(-1)=-8, so f is not one-to-one
(b) f(f(x)) = (x^2-9)^2-9
(c) y = x^3+3 <=> x^3 = y-3 <=> x = (y-3)^(1/3), so g^(-1)(x) = (x-3)^(1/3)
You can check that g(g^-1(x)) = [(x-3)^(1/3)]^3+3 = x-3+3 = 3 and g^(-1)(g(x))=(x^3+3-3)^(1/3)=x
2)
(a) I don't have your book
(b) i=1 to 3 ? it's not easy to read, then it's 5+5+5 = 15
3) (a)
AC+BC = (A+B)C
A+B=
0 0 0
0 3 0
1 0 1
AC+BC =
[0 0]
[-3 0]
[3 -1]
(b) 2A-3A = -A
So :
[1 1 -1]
[-2 -1 1]
[1 1 2]
(c)I don't have the definition, of the symbol
4)
(a) det(A)=2*4-2*2=4
A^(-1) = 1/4 *
[4 -1]
[-2 2]
A^(-1) =
[1 -1/2]
[-1/2 1/2]
(b) We don't have your definitions , but maybe it's AA^(-1) = I
So you can verify that AA^(-1) = I by doing the product
5) I don't have your method, I'll solve the conventional way with matrix :
A=
[1 1 1]
[2 -1 1]
[0 -1 1]
det(A)=1*(-1+1)-2*(1+1) = -4
The inverse using cramer formula (with determinants) is :
1/4*
[0 2 -2]
[2 -1 -1]
[2 -1 3]
The solution is Ab with b is the vertical vector (1 2 1)^T
(x1 x2 x3)^T = Ab = (1/2,-1/4,3/4)
6)/7) Can't do ...don't have your notes
8) It's wrong : (A-B)(A+B) = A*A-A*B-B*A-B^2 = A^2-AB-BA-B^2 (AB=BA is not true)
9) Can't do
Extra credits :
i)
We computed the inverse in previous exercise so :
So B =
[1 -1/2][3 0 ]
[-1/2 1/2][1 2 ]
B=
[5/2 -1]
[-1 1]
ii) Can't do
iii) Can't do