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). Check
<u,v>
=2u1v1+3u2v2+u3v3
is inner product space or not.
If yes assume
u= (8,0,-8) & v=
(8,3,16)
Find
||u||
||v||
|<u, v>|2
Unit vector in direction of u and v
Distance (u, v)
Angle between u and v
Orthogonal vectors of u and v.
(b). Show that
<u,v>
=u1v1-2u2v2+u3v3
Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4,
∥u + v∥ = 5. Find the inner product 〈u, v〉.
Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that
{a1, · · · ak} is a linearly independent set.
Suppose V is a finite dimensional inner product space. Prove that
every orthogonal operator on V , i.e., <T(u),T(v)> =
<u,v>, ∀u,v ∈ V , is an isomorphism.
Let U be a subset of a vector space V. Show that spanU is the
intersection of all the subspaces of V that contain U. What does
this say if U=∅? Need proof
1. For this question, we define the following vectors: u = (1,
2), v = (−2, 3).
(a) Sketch following vectors on the same set of axes. Make sure
to label your axes with a scale. i. 2u ii. −v iii. u + 2v iv. A
unit vector which is parallel to v
(b) Let w be the vector satisfying u + v + w = 0 (0 is the zero
vector). Draw a diagram showing the geometric relationship between...
2
Let u,v, and w be vectors, where u=(1,2,3,-1), v=(2,3,1,5) and
w=(3,5,4,4).
2.1
Construct a basis for the vector space spanned by u, v and w.
2.2
Show that c=(1,3,2,1) is not in the vector space spanned by the
above vectors u,v and w.
2.3
Show that d=(4,9,17,-11) is in the vector space spanned by the
above vectors u,v and w, by expressing d as a linear combination of
u,v and w.