Linear Algebra Answers

Suppose T€L(R^2) is defined by T(x, y) =(-3y,x).find the eigenvalues of T

Question 5:

(5.1)Assume that a vector a of length ||a|| = 3 units. In addition, a points in a direction that is 135◦ counterclockwise from the positive x-axis, and a vector b in th xy-plane has a length ||b|| = 1 3 and points in the positive y-direction.

(5.2) Find a ·b. Calculate the distance between the point (−1,√3) and the line 2x-2y-5=0

Question 6

Let u =< −2,1,−1,v =< −3,2,−1 >and w =< 1,3,5 >. Compute:

(6.1) u ×w,

(6.2)u ×(w ×v)and(u × w)×v.

Question 7

(7.1) (Find a point-normal form of the equation of the plane passing through P = (1,2,−3) and having n =< 2,−1,2 > as a normal.

(7.2) Determine in each case whether the given planes are parallel or perpendicular: (a) x +y +3z +10=0andx +2y −z =1,

(b)3x −2y +z −6=0and4x +2y −4z =0, (c)3x +y +z −1=0and−x +2y +z+3=0,

(d)x −3y +z+1=0and3x −4y +z−1=0.

Suppose T€ L(V ) is invertible.

(a) Suppose lemtha € F with lemtha not equal to 0. Prove that lemtha is an eigenvalue of T if and only if 1/lemtha is an eigenvalue of T^-1.

(b) Prove that T and T^-1 have the same eigenvectors.

Suppose V is the finite-dimensional and S; T€ L(V ). Prove that ST and T S have the same eigenvalues.

Check whether each of the following subsets of R 3 is linearly independent. i) {(1,2,3),(−1,1,2),(2,1,1)}. ii) {(3,1,2),(−1,−1,−3),(−4,−3,0)

Show that for any g element of L(V;C) and u element of V with g(u) not equal 0: V =null g operation { xu: x element of C}

let F^mxn be the set of all mxn matrices over the field F. is F^mxn is a vector space?

Suppose T€L(R^2) is defined by T(x, y) =(-3y,x).find the eigenvalues of T