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...
The Cauchy-Schwarz Inequality Let u and v be vectors in R 2
.
We wish to prove that -> (u · v)^ 2 ≤ |u|^
2 |v|^2 .
This inequality is called the Cauchy-Schwarz inequality and is
one of the most important inequalities in linear algebra.
One way to do this to use the angle relation of the dot product
(do it!). Another way is a bit longer, but can be considered an
application of optimization. First, assume that the...