Question
In: Math
The set R^2 with addition and scalar multiplication defined by (x1, y1) + (x2, y2) =...
The set R^2 with addition and scalar multiplication defined by
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
c(x1, y1) = (cx1, y1)
is not a vector space. Determine which axiom fails and find a counterexample that shows that it fails.
Solutions
milcah answered 2 years agoRelated Solutions
We define a relation ∼ on R^2 by (x1,y1)∼(x2,y2) if and only if (y2−y1) ∈ 2Z....
We define a relation ∼ on R^2 by (x1,y1)∼(x2,y2) if and only if
(y2−y1) ∈ 2Z. Show that the relation∼is an equivalence relation and
describe the equivalence class of the point (0,1).
1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1 −x2|+|y1 −y2|. (a) Prove...
1. Let ρ: R2 ×R2 →R be given by ρ((x1,y1),(x2,y2)) = |x1
−x2|+|y1 −y2|.
(a) Prove that (R2,ρ) is a metric space.
(b) In (R2,ρ), sketch the open ball with center (0,0) and radius
1. 2. Let {xn} be a sequence in a metric space (X,ρ). Prove that if
xn → a and xn → b for some a,b ∈ X, then a = b.
3. (Optional) Let (C[a,b],ρ) be the metric space discussed in
example 10.6 on page 344...
The parametric equations x = x1 + (x2 − x1)t, y = y1 + (y2 − y1)t...
The parametric equations
x = x1 +
(x2 −
x1)t, y
= y1 +
(y2 −
y1)t
where
0 ≤ t ≤ 1
describe the line segment that joins the points
P1(x1,
y1)
and
P2(x2,
y2).
Use a graphing device to draw the triangle with vertices
A(1, 1), B(4, 3), C(1, 6). Find the
parametrization, including endpoints, and sketch to check. (Enter
your answers as a comma-separated list of equations. Let x
and y be in terms of t.)
Show that the set ℝ2R2, equipped with operations (?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1) ? ⋅˜ (?,?)=(??+?−1,??−?+1) (1)defines a vector space...
Show that the set ℝ2R2, equipped with operations
(?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1)
? ⋅˜ (?,?)=(??+?−1,??−?+1)
(1)defines a vector space over ℝR.
(2)Show that the vector space ?V defined in question 1 is
isomorphic to ℝ2R2 equipped with its usual vector space operations.
This means you need to define an invertible linear map
?:?→ℝ2T:V→R2.
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 X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1 + X2 + X3, Y2...
Let X1,X2,X3 be i.i.d. N(0,1) random variables. Suppose Y1 = X1
+ X2 + X3, Y2 = X1 −X2, Y3 =X1 −X3. Find the joint pdf of Y =
(Y1,Y2,Y3)′ using : Multivariate normal distribution
properties.
Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent random samples, with...
Suppose that X1, X2, , Xm and Y1, Y2, , Yn are independent
random samples, with the variables Xi normally distributed with
mean μ1 and variance σ12 and the variables Yi normally distributed
with mean μ2 and variance σ22. The difference between the sample
means, X − Y, is then a linear combination of m + n normally
distributed random variables and, by this theorem, is itself
normally distributed.
(a) Find E(X − Y).
(b) Find V(X − Y).
(c)...
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