Prove or disprove: two consecutive rotations about two different
axis are commutative. That is, is RuRv...
Prove or disprove: two consecutive rotations about two different
axis are commutative. That is, is RuRv = RvRu? (Hint: For
simplicity, you can assume that the axis u is the x-axis and v is
the y-axis without loss of gnerality).
Prove or disprove: two consecutive rotations about two different
axis are commutative. That is, is RuRv = RvRu? (Hint: For
simplicity, you can assume that the axis u is the x-axis and v is
the y-axis without loss of generality).
Prove or disprove that the union of two subspaces is a subspace.
If it is not true, what is the smallest subspace containing the
union of the two subspaces.
prove or disprove
A Turing machine with two tapes is no more powerful than a
Turing machine with one tape. (That is, both types of machines can
compute the same set of functions.)
The cardinality of the set of irrational numbers is greater
than the cardinality of the set of all rational numbers.
The cardinality of the set of all algebraic numbers is exactly
the same as the cardinality of all real numbers.
Prove that all rotations and translations form a subgroup of the
group of all reflections and products of reflections in Euclidean
Geometry. What theorems do we use to show that this is a
subgroup?
I know that I need to show that the subset is
closed
identity is in the subset
every element in the subset has an inverse in the subset.
I don't have to prove associative property since that is already
proven with Isometries. What theorems for rotations...
Prove or disprove the statements: (a) If x is a real number such
that |x + 2| + |x| ≤ 1, then x 2 + 2x − 1 ≤ 2.
(b) If x is a real number such that |x + 2| + |x| ≤ 2, then x 2
+ 2x − 1 ≤ 2.
(c) If x is a real number such that |x + 2| + |x| ≤ 3, then x 2
+ 2x − 1 ≤ 2....
for unit quaternions used to represent rotations for 3D prove
that :
a) quaternion conjugate corresponds to rotation inverse (
Approach 1: negate the angle in the formula for converting from the
axis-angle form and apply trigonometric identities. Approach 2:
convincingly discusses what negating and axis description does in
terms of rotation effects and in terms of the formula)
b) negated quaternions represent the same rotation ( hint: add
2PI to the rotation angle in the axis angle formula, explain...
Topology
Prove or disprove ( with a counterexample)
(a) The continuous image of a Hausdorff space is Hausdorff.
(b) The continuous image of a connected space is
connected.