The set of all vectors in R 5 whose coordinates sum to zero forms a subspace. The following vectors are a generating set for the space. u1 = (2, −3, 4, −5, 2) u2 = (−6, 9, −12, 15, −6) u3 = (3, −2, 7, −9, 1) u4 = (2, −8, 2, −2, 6) u5 = (−1, 1, 2, 1 − 3) u6 = (0, −3, −18, 9, 12) u7 = (1, 0, −2, 3, −2) u8 = (2, −1, 1, −9, 7) (a) Find a basis in the collection above. (b) Let L = {(1, 1, 1, 1, −4),(1, −1, 3, −2, −1)}. Find 6 vectors in the collection, say H, such that L ∪ H spans the entire space.

Use the definition of absolute value and a proof by cases to prove that for all real numbers x, | − x + 2| = |x − 2|. (Note: Forget any previous intuitions you may have about absolute value; only use the rigorous definition of absolute value to prove this statement.)

Given two sets A,B prove A<---> B either using the definition of the schroeder-bernstein theorem

The Stone Company produces three sizes of window fans: small, medium and large. The operations manager has formulated the following LP model for fan production:

Maximize Profit Z

Z = 6×₁ + 8×₂ + 5×₃             (profit)

Subject to

                                3×₁ + 4×₂ + 5×₃ ≤ 160 hours (Labor)

                                1×₁ + 2×₂ + 3×₃ ≤ 100 pounds (Metal)

                                2×₁ + 2×₂ + 2×₃ ≤ 110 pounds (Plastic)

                                ×₃≥ 18 (Large Fan)

                                ×_1, ×_2, ×₃ ≥ 0

Briefly explain or define each of these parts of the model (1 point each):

a.             ×₁, ×₂ and ×₃.

b.             The 6 in the objective function.

c.             The product of the 8 and ×₂ in the objective function.

d.             The terms Labor, Metal, and Plastic.

e.             The product of 5 and ×3 in the labor constraint.

f.             The product of the 2 and ×₂ in the metal constraint.

g.           The 110 hours in the Right Hand Side (RHS) of the plastic constraint.

h.             ×₃ ≥ 18.

i.              ×₁, ×₂, ×₃ ≥ 0.

j.              What two key questions can be answered using this model?A8

Solve the following linear programming model graphically:

Max Z= 3x1 +4x2

Subject to: 2x1 + 4x2 <= 22

-x1 + 4x2 <= 10

4x1 – 2x2 <= 14 x1 – 3x2 <= 1

x1, x2, >=0

Clearly identify the feasible region, YOUR iso-profit line and the optimal solution (that is, d.v. values and O.F. Value.

Consider a system with the input/output relationship y(t) = x(t)cos(15πt).
(a) Is this system (i) linear, (ii) causal, (iii) stable, (iv) memoryless, (v) time-invariant, and (vi) invertible. Justify each answer with a clear mathematical argument. (b) Find the Fourier Transform Y (f) of y(t) in terms of the transform X(f) of x(t).

Repeat problem (2) for the system with the input-output relationship y(t) =R1 τ=0(1−τ)2x(t−τ)dτ.

A fluid flow is defined by u = (0.4x2 + 2t) m/s and v = (0.8x + 2y) m/s, where x and y are in meters and t is in seconds.

Part A

Determine the magnitude of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

V=

Part B

Determine the direction of the velocity of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θV, which the velocity makes with the x axis, measured counterclockwise from the positive x axis.

.

Part C

Determine the magnitude of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s.

Part D

Determine the direction of the acceleration of a particle passing through point x = 12.8 m, y = 1 m if it arrives when t = 3 s. Enter your answer as the angle θa, which the acceleration makes with the x axis, measured counterclockwise from the positive x axis.

Set up a spreadsheet that implements the secant method and then solves each of the problems. Use the graph of each function to select an initial guess. Recall the iteration formula for the secant method: X^k+1=x^k-[f(x^k)/f(x^k)-f(x^k-1)](x^k-x^k-1)

Put the formula for the function under the heading f(xk-1) and f(xk). In the cell under xk+1, put the secant method iteration formula. In the second row, replace the previous xk-1 with xk and then xk with xk+1. Now copy the two formulas down one row. At this point, one iteration of the secant method is displayed. To see more iterations, just copy the second row down for as many iterations as desired. If too many iterations are copied and the function difference becomes exactly zero, a divide by zero error will appear.

a. f(x) x-x^1/3-2

b.f(x)=xtanx-1

c.f(x)=x^4-e^x+1

d.f(x)x^2e^x-1

excel problem

Find the roots of the functions given using the bisection method. Use the graph of each function to choose points that bracket the root of interest.

a. f(x) x-x^1/3-2

b.f(x)=xtanx-1

c.f(x)=x^4-e^x+1

d.f(x)x^2e^x-1

TAM and its value and should we expect the use of mathematical models to hopefully provide more data/science-based justification?

A bipartite graph is drawn on a channel if the vertices of one partite set are placed on one line in the plane (in some order) and the vertices of the other partite set are placed on a line parallel to it and the edges are drawn as straight-line segments between them. Prove that a connected graph G can be drawn on a channel without edge crossings if and only if G is a caterpillar. (***Please do on paper)

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.