In the real vector space R 3, the vectors u1
=(1,0,0) and u2=(1,2,0) are known to lie in the span W of the
vectors w1 =(3,4,2), w2=(0,1,1), w3=(2,1,1) and w4=(1,0,2). Find
wi, wj ?{w1,w2,w3,w4} such that W = span({u1,u2,wk,wl}) where
{1,2,3,4}= {i,j,k,l}.
Let B equal the matrix below:
[{1,0,0},{0,3,2},{2,-2,-1}]
(1,0,0 is the first row of the matrix B)
(0,3,2 is the 2nd row of the matrix B
(2,-2,-1 is the third row of the matrix B.)
1) Determine the eigenvalues and associated eigenvectors of B.
State both the algebraic and geometric multiplicity of the
eigenvalues.
2) The matrix is defective. Nonetheless, find the general
solution to the system x’ = Bx. (x is a vector)
Find a pair of vectors, a → and b → that satisfy all of the
following conditions:
a → + b → = 〈 9 , 5 , 5 〉
a → is parallel to 〈 5 , 1 , 2 〉
b → is orthogonal (perpendicular) to {5,1,2}
#1 Let H= Span{v1,v2,v3,v4}. For each of the following sets of
vectors determine whether H is a line, plane ,or R3. Justify your
answers.
(a)v1= (1,2,−2),v2= (7,−7,−7),v3= (16,−12,−16),v4= (0,−3,−3)
(b)v1= (2,2,2),v2= (6,6,5),v3= (−16,−16,−14),v4= (28,28,24)
(c)v1= (−1,3,−3),v2= (0,0,0),v3= (−2,6,−6),v4= (−3,9,−9)
#2 Plot the linesL1: x= t[4−1] and L2: x= [−4−2] + t[4−1] using
their vector forms. If[12k]is onL2. What is the value of k?
A) Create two vectors in R3 ( 3D vectors) such that they are
orthogonal.
b) Using the two vectors from a) above, determine the cross
product of the two vectors
c)Is it possible to write the vector [0,0,1] using scalar
multiples of these vectors?