Question
In: Math
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 additive inverse −u for u = (u1, u2). Note: is not (−u1, −u2), so don’t write that.
3)Show that V is not a vector space.
Solutions
milcah answered 2 years agoRelated Solutions
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 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.
Let ?V be the set of vectors in ?2R2 with the following definition of addition and...
Let ?V be the set of vectors in ?2R2 with the following
definition of addition and scalar multiplication:
Addition: [?1?2]⊕[?1?2]=[0?2+?2][x1x2]⊕[y1y2]=[0x2+y2]
Scalar Multiplication: ?⊙[?1?2]=[0??2]α⊙[x1x2]=[0αx2]
Determine which of the Vector Space Axioms are satisfied.
A1. ?⊕?=?⊕?x⊕y=y⊕x for any ?x and ?y in ?V
? YES NO
A2. (?⊕?)⊕?=?⊕(?⊕?)(x⊕y)⊕z=x⊕(y⊕z) for any ?,?x,y and ?z in
?V
? YES NO
A3. There exists an element 00 in ?V such that ?⊕0=?x⊕0=x for
each ?∈?x∈V
? YES NO
A4. For each ?∈?x∈V, there exists an...
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
Let A = {a1,...,an} be a set of real numbers such that ai >= 1 for...
Let A = {a1,...,an} be a set of real numbers such that ai >=
1 for all i, and let I be an open interval of length 1. Use
Sperner’s Theorem to prove an upper bound for the number of subsets
of A whose elements sum to a number inside the interval I.
6. Let R be a relation on Z x Z such that for all ordered pairs...
6. Let R be a relation on Z x Z such that for all ordered pairs
(a, b),(c, d) ∈ Z x Z, (a, b) R (c, d) ⇔ a ≤ c and b|d . Prove that
R is a partial order relation.
Consider the set of ordered pairs shown below. Assuming that the regression equation is y with...
Consider the set of ordered pairs shown below. Assuming that the
regression equation is y with caret=1.087+0.724x and the
SSE=5.362, construct a 90% prediction interval for x=1.
x 6 1 4 4 4 1
y 6 2 5 4 2 2
Calculate the upper and lower limits of the prediction
interval.
UPL=
LPL=
Consider the set of ordered pairs shown below. Assuming that the regression equation is y with...
Consider the set of ordered pairs shown below. Assuming that the
regression equation is y with caret=5.322-0.029x and the
SSE=18.794, construct a 95% prediction interval for x=6.
x 4 6 3 3 5
------------------
y 6 7 4 7 2
Click the icon to view a portion of the student's
t-distribution table. Calculate the upper and lower limits of the
prediction interval.
UPL=
LPL=
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...
Consider the following set of ordered pairs. Complete parts a through c. x 1 1 3...
Consider the following set of ordered pairs. Complete parts a
through c.
x
1
1
3
3
1
y
−1
1
4
3
−1
a. Calculate the slope and y-intercept for these data.
^y=______ + (______) x
(Round to four decimal places as needed.)
b. Calculate the total sum of squares (SST).
SST=
(Round to one decimal place as needed.)
c. Partition the sum of squares into the SSR and SSE.
SSE=
(Round to three decimal places as needed.)...
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