Vectors and Linear Algebra

Compute the following vector quantities: a. (i-j-k)x(2i+3j-k) b. (3i-j-k).[(i-j)x(i+j-k)]

Relations between Non-Parallel Planes

Let P1 and P2 be two dimensional planes that are not parallel. If these planes are contained in R3 then they must intersect in a line. Prove that if they are contained in R4 instead then they can intersect either along a line, or at a single point. (HINT)- A two dimensional plane in R4 is determined by two equations in four unkn ...continues

Linear Alegbra : Systems of Equations

Find the solution space of the following system of equations. x1-x2+4x3-x4=2 2x1 -2x3+4x4=4 2x1-x2+3x3+x4=4 x2-5x3+2x4=0

Linear Alegbra : Vector Space

Let V= (x,y) in R2{y=3x+1} with addition and multiplication by a scalar defined on V by: (x,y)+ (x',y')= (x+x',y+y'-1) k(x,y)=(kx,k(y-1)+1) Given that with these definitions, V satisfies vector space axioms 1,2,3,6,8,9,and 10 determine whether or not V is a vector space by checking to see if axioms 4,5,7,are also satisfied.

Vectors

Suppose that a "skew" product of vectors in R2 is defined by (u,v)=u1v1-u2v2 Prove that (u,v)squared >equal too (u,u)(v,v). (NOTE; This is just the reverse of the Cauchy- Schwartz inequality for the ordinary dot product.)

Linear alegbra and vectors

Use the geometric method of linear programming to maximize the objective function f(x,y)=3x-6y subject to the constraints. x>= 1 x-y<= 3 2x+y>= 6 2x+y<= 8

Equation of the plane: 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)