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# Linear Algebra

Studying linear algebra means studying linear equations, linear maps, and how they are shown through vector spaces and matrices. From geometry to functional analysis, linear algebra is key to many parts of mathematics.

Geometry was one of the first key uses of linear algebra, beginning with Cartesian geometry, but linear algebra now has a strong relationship with science and engineering because of its ability to model different phenomena.

### Region Infected by Parasite Exponential Decay Functions

Question 1 The population of honey bees in a region is infected by a parasite that kills off bees, so that the number of bees decreases with time. Numbers of bees are usually measured by the number of 'colonies' (groups of bees living together) since it is very difficult to measure the exact number of bees (there m

### Vector Space and Subspace

Define vectors pace and subspace with examples. State and prove a necessary and sufficient condition for a subset of vectors to be a subspace. Show that the intersection and union of two sub spaces are also sub spaces.

### Algebra for Value Equations

Please help me solve this problem. Given the equation: y = |2x - 1| + 1 Absolute value equations involving linear powers of x result in two lines which cross each other. 1. What are the equations of the two lines from the absolute value equation? y1 = m1x1 + b1 and y2 = m2x2 + b2 2. Plot these two lines on a piece of

### Trigonometry: Surface Area vs. Volume Flashcards

Hello! I was in the hospital during my spring break, and I am in desperate need of "flash cards" covering surface area as well as volume for typical polygons. Pythagorean theorem, cylinders, cones, etc. Please help! Best regards, an aspiring applied anthropologist

### Inner Product and Orthogonal Vectors

3. Let V be an R-vector space with inner product ( - . - ). (a) Let S = {b1, b2, ...} be a set of vectors in V. Define what it means for S to be an orthogonal set or an orthonormal set with respect to the inner product. (b) Let V = R^4 and let ( - . - ) be the dot product. Apply the Gram-Schmidt orthoganlisation process to t

### Using the Cayley-Hamilton theorem

See attached file for the matrix. (a) Calculate the characteristic polynomial Pa(t). Is a diagonalisable? (b) State the Cayley-Hamilton theorem for a square matrix a. (c) Using the Cayley-Hamilton theorem, compute a^5 with no more than one explitic matrix multiplication.

### Linear transformation in Matrix form

Let P3 denote the real vector space of polynomial functions of degree up to 3, i.e. p3-{f(x)=a3x^3+a2x^2+a1x| aiER} Consider the linear transformation D: P3--> P3 given by the derivative D(f) = d/dx f a) What is the kernel of D? Give a basis of the kernel b) The set B={1, x, x^2, x^3} forms a basis of D. Write the matrix

### Linear Algebra: Vectors and Least Squares Problems

Please see the attached document for formatted version of the problem. 1. Consider the set of vectors: V1 = [6, 0, 2]. V2 = [2, -1, 1], V3 = [-4, -1, -1] Do they form a basis in R^2? Give reasons. What is the dimension of space spanned by these vectors? 2. Find the orthogonal projection of the vector u = (-2,

### Height vs Arm span Algebra Scatter Plots

Project Option 1â€”Individually There are many measurements of the human body that are positively correlated. For example, the length of one's forearm (measured from elbow to wrist) is approximately the same length as the foot (measured from heel to toe). They are positively correlated because, as one measurement increases, so

### Questions regarding vector spaces

9. Let u, x, y, z be elements in a vector space V where u = 3x - 4y +5z, v = 4x + 5y - 6z, and 2u + 3v = x + y + z Show that {x,y,z} is a linearly dependent set. 10. Find a basis for the space spanned by the following vectors (see attached file) 11. Let W = {p epsilon P2 | p'(1) = 0}. That is, W are the functions

### Vector Subspace Example

Calculate the dimension of the subsapce of P4(t) (polynomials of degree at most 4 in the indeterminate t) consisting of polynomials p(t) E P4(t) satisfying p(0) = p(1) Please show complete steps to this vector subspace dimensions question.

### Vandermonde Determinant Properties

Please show complete steps for the question on Vandermonde determinant properties attached. Let x1, c2, x3, and x4 be real numbers. (i) Compute the determinant of [1 x1 x1^2 x1^3; 1 x2 x2^2 x3^3; 1 x3 x3^2 x3^3; 1 x4 x4^2 x4^3] (ii) Explain briefly and concisely while the given matrix is nonsingular (invertible, or ha

### Diagonalization of Linear Operator

Consider the linear operator T:R^3 given by T (see attached) Determine the eigenvectors and the corresponding eigenvalues of T. If T diagonalizable? Why or why not?

### Partial Fraction Decomposition Case

Let a1, a2,..., an be n distinct numbers and set f(x)=see attached. An identity see attached is called a partial fraction decomposition of f(x). i. Show that the preceding identity is equivalent to a nonhomogeneous system of n linear equations in the variable c1, c2,...,cn ii. Show that the system of homogeneous equations

### Scheduling Optimization on Resource Planning

11. Gerald Glynn manages the Michaels Distribution Center. After careful examination of his database information, he has determined the daily requirements for part-time loading dock personnel. The distribution center operates 7 days a week, and the daily part-time staffing requirements are: Day M T W Th F S Su Requirements

### 2Q

1. Diagonalize the matrix A for A=[-4 -1 ] [5 2 ] 2. Suppose the y'=Ay for the given A. Find a solution for the system of ordinary differential equations with initial condition y(0)=[0] [2] A=[-3 1] [ 1 -3]

### Regression Analysis: Sales Total Vs Profit Total

A company produces the financial results shown in the table below. The executives at the firm have good reason to believe that \$10 million in sales will be generated in 2010. Using simple linear regression, you advise them that this will equate to... Year Sales Totals (in millions) Profit Totals (in millions) 1998 \$7.0 \$0.

### Question on Linear Equations setup

1. Juanita sells two different computer models. For each Model A computer sold she makes \$45, and for each Model B computer sold she makes \$65. Juanita set a monthly goal of earning at least \$4000. A) Write a linear inequality that describes Juanita's options for making her sales goal. 2. The Candy Shack sells a particular

### Linear Transformations, Matrices, Orthogonal Projections

R stands for the field of real numbers. C stands for the field of complex numbers. 1. Let T be a linear transformation from the set P2(R) of all polynomials of degree at most 2 into itself. T: P2(R) --> P2(R), given by T(f) = f' - f'', fEP2(R), where f' is the first and f'' is the second derivative of f. (a) Find the null

### Linear Algebra - drying time

The drying time of varnish depends on the amount of certain chemical that s added. a. Determine a best (least-squares) ft parabola of the form: T(m) = x1 + x2m + x3m^2 to the data provided n the table below b) Also, estimate the drying time of the varnish when 3.5 grams of the chemical are added. Mass (m) of Additive (Gr

### Using a Graphic Method to solve Linear Programming

The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of d

### Using algebraic thinking for problem-solving

1. Do the problem "Tables and Seats" in this topic. As you work on the problem, see if you can arrive at your solution from several different approaches. Upload your completed solution and methods. 2.Reflect on the aspects of algebraic thinking you discovered in the problem that were applicable to your problem-solving strateg

### Non-Empty Subsets of a Group

Let G be a group and H be a non-empty subset of G. We define H^2 as the set of elements of G which can be written in the form h_1 h_2, with h_1 ,h_2âˆˆH. Let H be finite. Prove that H is a subgroup if and only if H^2=H.

### Write formal proofs/Logic laws for proofs

See the attachment for all questions. 18. Given: If you are wealthy, then you are a success. You are wealthy or you are. healthy. You are not healthy. Let W represent: "You are wealthy." Let S represent: "You are a success." Let H represent: "You are healthy." Prove: You are a success. 19. Given: The object i

### The solution gives detailed steps on Constructing Arguments and Reasoning for a theorem proof.

Introduction: This task is focused on developing a logical progression of steps to develop a reasonable and complete geometric proof. To open your assigned proof, download and open the attached "Proof Problem Generator" and insert your initials into the appropriate box. Then follow the link to your assigned proof that you will

### Determine the condition of a system of equations

Which of the following can be solved by using a system of equations? A) Laura and Mark went to the movies and spent \$29 on movie tickets and snacks. Mark paid \$10 more than Laura. How much did each spend? B) Laura and Mark went to the movies and spent \$29 on movie tickets and snacks. Popcorn cost \$5. How much were the ti

### Logic Application: Guess Your Card

Necessary Background The following project uses the game of Guess Your Card. This is a game in which Each player draws (without looking) three (3) cards. Each card has a number between 1 and 9 on it. The players then place their cards on their heads so that everyone but the players can see the cards. The object of the gam

### Linear Correlation, Critical Value, and Regression Equations

Find the value of the linear correlation coefficient r, the critical value, and determine whether there is a linear correlation. 1) The paired data below consist of the costs of advertising (in thousands of dollars) and the number of products sold (in thousands): (see attached file for data) Use the given data to find the

### Determining Slope And Intercept of A Line

Find the slope and y intercept of the line. Graph the line 7x-5y=35 Decide whether the given linear equations are parallel, perpendicular or neither. y=4x+9 y=-1/4x+5/4 Decide whether the given linear equations are parallel, perpendicular or neither. y=5x-7 y=5x+2 Write the standard from of the equation and the general

### Congruence with Incongruent Solutions

Let n be an integer greater than 2. For which values of n (if any) does the congruence 12x â‰¡ 8 (mod n) have exactly two incongruent solutions (mod n)? Justify your answer.