Proof:

Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.

1----

. Is the statement​ "Elementary row operations on an augmented matrix never change the solution set of the associated linear​ system" true or​ false? Explain.

A.

​True, because elementary row operations are always applied to an augmented matrix after the solution has been found.

B.

​False, because the elementary row operations make a system inconsistent.

C.

​True, because the elementary row operations replace a system with an equivalent system.

D.

​False, because the elementary row operations augment the number of rows and columns of a matrix.

2---

Indicate whether the statements given in parts​ (a) through​ (d) are true or false and justify the answer.

a. Is the statement​ "Every elementary row operation is​ reversible" true or​ false? Explain.

A.

​False, because only interchanging is a reversible row operation.

B.

​True, because interchanging can be reversed by​ scaling, and scaling can be reversed by replacement.

C.

​False, because only scaling and interchanging are reversible row operations.

D.

​True, because​ replacement, interchanging, and scaling are all reversible.

3----

In parts ​(a) through​ (e) below, mark the statement True or False. Justify each answer.

a. The echelon form of a matrix is unique. Choose the correct answer below.

A.

The statement is true. The echelon form of a matrix is always​ unique, but the reduced echelon form of a matrix might not be unique.

B.

The statement is true. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.

C.

The statement is false. The echelon form of a matrix is not​ unique, but the reduced echelon form is unique.

D.

The statement is false. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B + CX) ?1 = C Write an expression for the matrix X in terms of A, B, and C. You may assume invertibility of any matrix when necessary.

(b) Suppose D is a 3 × 5 matrix, E is a 5 × c matrix, and F is a 4 × d matrix. Find the values of c and d for which the statement “det(DEF) = 1” can be valid. Explain your answer.

(c) Find all (real or complex) values of x such that the matrix GH is invertible, where G =

x^2, ?1

x , x ? 2

H = x ? 1 , ?2

1 , x + 1

An investment club has set a goal of earning 15% on the money they invest in stocks. The members are considering purchasing three possible stocks, with their cost per share (in dollars) and their projected growth per share (in dollars) summarized in the table. (Let x = computer shares, y = utility shares, and z = retail shares.)
Stocks
Computer (x)   Utility (y)   Retail (z)
Cost/share   30   44   26
Growth/share   6.00   6.00   2.40
(a) If they have $392,000 to invest, how many shares of each stock should they buy to meet their goal? (If there are infinitely many solutions, express your answers in terms of z as in Example 3.)

b) If they buy 1500 shares of retail stock, how many shares of the other stocks do they buy?
computer

c)What if they buy 3000 shares of retail stock?
computer
utility

(c) What is the minimum number of shares of computer stock they will buy?

D)What is the number of shares of the other stocks in this case?
utility
retail

(d) What is the maximum number of shares of computer stock purchased?

E)What is the number of shares of the other stocks in this case?
utility
retail

Given two sets S and T, the direct product of S and T is the set of ordered pairs S × T = {(s, t)|s ∈ S, t ∈ T}.Let V, W be two vector spaces over F.

(a) Prove that V × W is a vector space over F under componentwise addition and scalar multiplication (i.e. if (v1, w1),(v2, w2) ∈ V × W, then (v1, w1) + (v2, w2) = (v1+w1, v2+w2) and a(v, w) = (av, aw) for any (v, w) ∈ V ×W, a ∈ F).

(b) If dim V = n and dim W = m, prove that dim V × W = n + m by constructing a basis.

1.     Describe your personal view on algebra being a requirement for your degree program. How do you feel about being forced to take math?

2.     Mathematics is part of the liberal arts curriculum, as well as the curriculum for any STEM field. Describe what you think of when you hear the phrase “liberal arts.” What about “STEM?”

3.     Describe what you think the purpose of higher education is.

When Gustavo and Serrana bought their home, they had a 5.1% loan with monthly payments of $870.60 for 30 years. After making 78 monthly payments, they plan to refinance for an amount that includes an additional $35,000 to remodel their kitchen. They can refinance at 4.5% compounded monthly for 25 years with refinancing costs of $625 included with the amount refinanced.

A) Find the amount refinanced. (Round your answer to the nearest cent.)

(b) Find their new monthly payment. (Round your answer to the nearest cent.)

(c) How long will it take to pay off this new loan if they pay $1200 each month? (Round your answer up to the next whole number.)
payments

Type in only your numerical answer to the following problem; do not type any words or letters with your answer. The population of Neverland was 2.3 billion, one hundred years ago. Currently the population is 3.2 billion. What will this population be 100 years from now? NOTE: Round to the nearest tenth of a billion.

The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $170; the quantity demanded each month is 750 when the unit price is $150. The suppliers will market 750espresso makers if the unit price is $110. At a unit price of $130, they are willing to market 2250 units. Both the demand and supply equations are known to be linear.

(a) Find the demand equation.
p =

(b) Find the supply equation.
p =

(c) Find the equilibrium quantity and the equilibrium price.

A group of retailers will buy 120 televisions from a wholesaler if the price is $375 and 160 if the price is $325. The wholesaler is willing to supply 88 if the price is $320 and 168 if the price is $410. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market. Find (q,p).

The daily demand for ice cream cones at a price of $1.20 per cone is 50 cones. At a price of $2.20 per cone, the demand is 30 cones. Use linear interpolation to estimate the demand at a price of $1.50 per cone.

The table shows the estimated number of E. coli bacteria in a lab dish t minutes after the start of an experiment.

A. Using t as the independent variable, find the model that best fits the data. Round values to the nearest thousandths.

B. How long does it take the population of E. coli to triple?

We can approximate the continuous-time tank model of the previous problem by a discrete model as follows.

Assume that we only observe the tank contents each minute (time is now discrete). During each minute, 20 liters (or 10% of each tank’s contents) are transferred to the other tank.

Let x1(t) and x2(t) be the amounts of salt in each tank at time t. We then have:

x1(t + 1) = 9 /10 x1(t) + 1 /10 x2(t)

x2(t + 1) = 1 /10 x1(t) + 9 /10 x2(t)

Formulate the problem in the form x(t + 1) = Ax(t) where A is a 2 × 2 matrix, then solve for the amount of salt in each tank as a function of time using the eigenvalues and eigenvectors of A.

Sketch the graphs of the amount of salt in each tank as functions of time.

How does your solution compare to the continuous time model?