Question
In: Advanced Math
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and...
Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on S by:
α(x1,x2)=(αx1,αx2)
(x1,x2)⊕(y1,y2)=(x1 +y1,0)
We use the symbol⊕to denote the addition operation for this system in order to avoid confusion with the usual addition x+y of row vectors.
Show that S, together with the ordinary scalar multiplication and the addition operation⊕, is not a vector space.
Test ALL of the eight axioms and report which axioms fail to hold.
Solutions
Colby Messinger answered 2 years agoRelated Solutions
Let V be the set of all ordered pairs of real numbers. Consider the following addition...
Let V be the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
Show that V is not a vector space.
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the...
Let V be the set of all ordered triples of real numbers. For u = (u1,...
Let V be the set of all ordered triples of real numbers. For u =
(u1, u2, u3) and v = (v1, v2, v3), we define the following
operations of addition and scalar multiplication on V :
u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2,
ku3).
For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then
u +...
Let S be a set of n numbers. Let X be the set of all subsets...
Let S be a set of n numbers. Let X be
the set of all subsets of S of size k, and let
Y be the set of all ordered k-tuples
(s1, s2, ,
sk)
such that
s1 < s2
< < sk.
That is,
X
=
{{s1, s2, ,
sk} | si S and all si's
are distinct}, and
Y
=
{(s1, s2, ,
sk) | si S and s1 <
s2 < < sk}.
(a) Define a one-to-one correspondence
f : X → Y.
Explain...
1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...
1: Let X be the set of all ordered
triples of 0’s and 1’s. Show that X consists of 8 elements
and that a metric d on X can be defined by
∀x,y ∈ X:
d(x,y) := Number of places where
x and y have different entries.
2: Show that the non-negativity of a metric can
be deduced from only Axioms (M2), (M3), and (M4).
3: Let (X,d) be a metric
space. Show that another metric D on X can...
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not identified. V = R^2 , < X1 , X2 > + < Y1 , Y2
> = < X1 + Y1 , 0> c< X1 , X2 >...
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = { (x1, x2, x3) ∈ R^3 | x1 > or equal to 0,
x2 > or equal to 0, x3 > or equal to 0}
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = { f : R --> R | f(1) = 0 }
Determine whether the set with the definition of addition of vectors and scalar multiplication is a...
Determine whether the set with the definition of addition of
vectors and scalar multiplication is a vector space. If it is,
demonstrate algebraically that it satisfies the 8 vector axioms. If
it's not, identify and show algebraically every axioms which is
violated. Assume the usual addition and scalar multiplication if
it's not defined. V = {all polynomials with real coefficients with
degree > or equal to 3 and the zero polynomial}
Let Rx denote the group of nonzero real numbers under multiplication and let R+ denote the...
Let Rx denote the group of nonzero real numbers under
multiplication and let R+ denote the group of positive
real numbers under multiplication. Let H be the subgroup {1, −1} of
Rx. Prove that Rx ≈ R+ ⊕ H.
Let A be a set of real numbers. We say that A is an open set...
Let A be a set of real numbers. We say that A is an open set if
for every x0 ∈ A there is some δ > 0 (which might
depend on x0) such that (x0 − δ,
x0 + δ) ⊆ A. Show that a set B of real numbers is closed
if and only if B is the complement of some open set A
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