consider the vectors:
v1=(1,1,1)
v2=(2,-1,1)
v3=(3,0,2)
v4=(6,0,4)
a)find the dimension and a basis
W=Span(v1,v2,v3,v4)
b) Does the vector v=(3,3,1) belong to W. Justify your answer
c) Is it true that W=Span(v3,v4)? Justify your answer
Find a basis and the dimension of W. Show algebraically how you
found your answer.
a. W = {(x1, x2, x3, x4) ∈ R^4 | x2 = x3 and x1 + x4 = 0}
b. W = {( A ∈ M 3x3 (R) | A is an upper triangular matrix}
c. W = { f ∈ P3 (R) | f(0) = 0.
V is a subspace of inner-product space R3, generated
by vector
u =[1 1 2]T and v
=[ 2 2 3]T.
T is transpose
(1) Find its orthogonal complement space V┴ ;
(2) Find the dimension of space W = V+ V┴;
(3) Find the angle q between u and
v; also the angle b between
u and normalized x with respect
to its 2-norm.
(4) Considering v’ =
av, a is a scaler, show the
angle q’ between u...
(a) Show that the lines
r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s) = (1,0,0) +
s(0,1,−2)
are skew.
(b) The two lines in (a) lie in parallel planes. Find equations for
these two planes. Express your
answer in the form ax+by+cz +d = 0. [Hint: The two planes will
share a normal vector n. How would one find n?]
would one find n?]
A = [4, 5, 9]
B = [-4, 5, -7]
C = [2, -7, -8, 5]
D = [1, -9, 5, -3]
E = [3, 3, -1]
Uz = 1/|z| ^z
d(X,Y) = (Rθ) d = diameter R = Radius θ = Theta
Find
a. Uc
b. d (D, C)
c. Let P = B + 3E, UP =
d. A x B
e. 3B x E
f. C x D