Linear alegbra and vectors

Find equations for the indicated geometrical objects. The plane which contains the line of intersection of the planes: x-2y-z=1 x+y+2z=4 and the point P: (1,2,-1)

Linear Alegbra : Vectors

Find equations for the indictated geometrical objects The line through the point P=(1,1,1) and perpendicualar to the plane 4x-2y+6z=3

This problem asks the student to find the eigenvalues of a 3x3 matrix.

Find the eigenvalues of the following matrix Q = Mat[0 0 -2; 1 2 1; 1 0 1]. (See attached file for clearer version.)

Permutations

Please see the attached file for the fully formatted problem. Find the order of sigma^1000 , where sigma is the permutation (123456789). (378945216)

Normal Subgroups

Please see the attached file for the fully formatted problem. Let G be a group and let D ={(a,a,a):a E G}. Prove that D is a normal subgroup of G+G+G if and only if G is Abelian.

Abelian Groups

Find examples of the following. Explain your answers. (a) A nonabelian group G and a proper normal subgroup S such that G/S is cyclic.

Kernel of Homomorphism

Please see the attached file for the fully formatted problems. Suppose  is an onto homomorphism from ℤ16 to a group G of order 4. Find ker(). Explain your answer.

Element of Finite Order

Prove that every element of Q/Z has finite order, where Q is the set of rational numbers with group operation + and Z is the set of integers.

Radioactive isotope

Complete the table for the given radioactive isotope. Isotope 226^Ra Half-life(years) 1620 Initial Quantity ______ Amount after 1000 yrs. 1.5g

Exponential and Logarithmic Equations : Modeling

The number of trees per acre N of a certain species is approximated by the model N=68(10^-0.04x), 5 <_ x <_ 40 (<_ = less than or = to) Where x is the average diameter of the trees (in inches) three feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when N=21.