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Linear Algebra

Normal Subgroups

Find all maximal normal subgroups of Z[p] × Z[q], where p and q are relatively prime. Would the elements from Z[p] have to be one that are relatively prime to q and vice versa?

Radical Expressions and Linear Equations and Inequalities

14. 5x(x + 1)(x - 1) > 0 24. t(x) = - t(4), t(4), t(0), t(-1), and t(- ) 28. Find the domain of the function t in exercise 24. Graph. 42. f(x) = Simplify. 56. - For the given function, find the indicated function values. 60. g(x) = - g(-62), g(0), g(-13), an

System of equations

4. The St. Marks community bbq served 250 dinners. a child's plate cost 3.50 and an adult's plate cost 7.00. A total of 1347.50 was collected. how many of each type of plate was served? 8. Deep thought granola is 25% nuts and dried fruit. Oat dream granola is 10% nuts and dried fruit. How much of deep thought and how much of

Rings : Annihilators

(5) Let R be a ring with 1 and M a left R-module. If N is a submodule of M, the annihilator of N in R is defined to be: {r in R/rn=0 for all n in N} Prove that the annihilator of N in R is a two-sided ideal of R.

Vectors : Linear Transformations, Eigenvectors and Eigenvalues

Given the plane (x, -y, 0). This is the plane that is parallel with the Z axis and intersects the x,y plane through the line x-y=0 in 3 space do the following: a) show the normal vector b) construct a matrix that will reflect points across this plane c) Compute the eigenvalues for this matrix d) compute the eig

Conjugacy Classes

Let K={k1,....km} be a conjugacy class in the finite group G. a) Prove that the element K=k1+k2+....km is the center of the group ring R[G] (check that g^-1Kg=K for all gin G) b) Let K1,....Kr be the conjugacy classes of G and for each Ki let Ki be the element of R[G] that is the sum of the members of Ki. Prove that an el

Solve each system by substitution

Solve each system by substitution y = 2x -8 4x + 3y = 1 solve each system by the addition method 3x + 2y = 3 4x -3y + - 13 Determine whether each system is independent, inconsistent or dependent y = 3x -5 y = 3x + 2 y = 2x -3 y = 5x - 14 Solve the following system by the elimination of variables x +

Systems of Equations

Find a linear function f(x) = mx + b whose graph has the given slope and y-intercept. 6. Slope: 4/5 ; y-intercept: (0,28) Find an equation of the line having the given slope and containing the given point 14. m = 3, (-2, -2) Find an equation of the line containing the given pair of points. 28. (-4, -7) and (-2, -1

Solve for F

See attached file for full problem description. When converting from Fahrenheit degrees to Celsius degrees, a well known formula is used: C= 5(F - 32) / 9

Coordinate system

See attached file for full problem description. 1. Find the slope of the line passing through the points (-8, -3) and (-2, 2). A) B) C) D) 2. Give the coordinates of the point graphed below. A) (4, 0) B) (-4, 0) C) (0, 4) D) (0, -4) 3. Find the slope of the graphed line.

Normal Subgroups, Centralizers and Semi-Direct Products

1. I f A and B are normal subgroups of G such that G/A and G/B are abelian, prove that G/(A intersect B) is abelian 2. Let H and K be groups, let f be a homomorphism from K into Aut(H) and as usual identify H and K as subgroups of G= H x_f K( x_f denotes product of H and K under f). Prove that C_K(H)= Ker(f) ps. C_K(H) is

Argue why every odd order subgroup has to be a subgroup of the cyclic odd order subgroup K of index 2, and by the divisibility argument it still could be a subgroup of G and not to be contained entirely in K.

Please see the attached file first. I did the proof, but it's weak, since I can't find a way to argue why every odd order subgroup has to be a subgroup of the cyclic odd order subgroup K of index 2, and by the divisibility argument it still could be a subgroup of G and not to be contained entirely in K.