Question

Consider the set V = (x,y) x,y ∈ R with the following two operations: • Addition: (x1,y1)+(x2,y2)=(x1 +x2 +1, y1 +y2 +1) • Scalarmultiplication:a(x,y)=(ax+a−1, ay+a−1). Prove or disprove: With these operations, V is a vector space over R

Answer #1

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 = R^2 , < X1 , X2 > + < Y1 , Y2 > =
< X1 + X2 , Y1 +Y2> c< X1 , X2...

Q 1 Determine whether the following are real vector spaces.
a) The set C with the usual addition of complex numbers and
multiplication by R ⊂ C.
b) The set R2 with the two operations + and · defined
by (x1, y1) + (x2, y2)
= (x1 + x2 + 1, y1 + y2
+ 1), r · (x1, y1) = (rx1,
ry1)

Let V=R2 with the standard scalar multiplication and nonstandard
addition given as follows: (x1, y1)⊕(x2, y2) := (x1x2, y1+y2). Show
that (V,⊕, .) is not a vector space.

Are the following vector space and why?
1.The set V of all ordered pairs (x, y) with the addition of
R2, but scalar multiplication a(x, y) = (x, y) for all a
in R.
2. The set V of all 2 × 2 matrices whose entries sum to 0;
operations of M22.

Prove or disprove: GL2(R), the set of invertible 2x2 matrices,
with operations of matrix addition and matrix multiplication is a
ring.
Prove or disprove: (Z5,+, .), the set of invertible
2x2 matrices, with operations of matrix addition and matrix
multiplication is a ring.

Exercise 9.1.11 Consider the set of all vectors in R2,(x, y)
such that x + y ≥ 0. Let the vector space operations be the usual
ones. Is this a vector space? Is it a subspace of R2?
Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy =
0. Is this a subspace of R2? Is it a vector space? The addition and
scalar multiplication are the usual operations.

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).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
Show that V is not a vector space.

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).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• 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:...

Complete the following fact. Suppose X,Y are independent RVs and
x1 < x2 and y1 < y2 are real numbers. Then P(x1 <_?_≤ x2,
_?__ <Y≤y2)=P(x1<X≤ _?_ )(y1<+_?_ ≤y2). Please fill in
question marks.

Consider a two-good economy c = (c1, c2) where the goods can
only be consumed in positive integer choices, that is c ∈ Z^2 and c
≥ 0. Consider the following three consumption bundles, x = (2,1), y
= (α, 2), z = (2, β).. These are the only three consumption bundles
Anne can choose from. Anne’s preferences are such that x ≻ y and y
≻ z, where “≻” means strict preference. Anne’s preferences are
complete and satisfy transitivity...

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