(a) Given the metric space R2, d > where d is the usual metric defined on R2. Let S CR²be a subset defined by{(x, y) = R²: x² + y² <1, x² + (y − 2)² ≤ 4}(i) Is the set S relatively open or relatively closed in subspace that is the open ballB₁ (0,0)? Justify.(ii) Is the set S relatively open or relatively closed in subspace that is the closed ballB₂(0, 2)? Justify your answer.

Answered: Image /qna-images/answer/9d21adb9-a655-4747-a89c-795192fb985a.jpg

Answered: (a) Given the metric space R², d where…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 38E: Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1)...

See similar textbooks

Similar questions

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].
Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Which vector spaces are isomorphic to R6? a M2,3 b P6 c C[0,6] d M6,1 e P5 f C[3,3] g {(x1,x2,x3,0,x5,x6,x7):xiisarealnumber}
Q4) Consider the space (X.T) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).
Show that V={(x,2020x): x in R} is a vector space
4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a subset of RN. (d) C*(RN). (e) C∞ (RN).
H.W: Let V = R2, defined the addition by: (U1 +U2) + (V1 + V2)= ( U1 +V1 , Uz +V2 + 1) With standard scalar multiplication. Is V a vector space justify ? Н.W: Let V = {(x, y):x > 0} with standard operation on R2. Is V a vector %3D space?
Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map f: EF, the following properties are = n. equivalent: (a) f is bijective. (b) f is surjective. (c) f is injective.. (d) Ker f = (0).
Is the set V = {(x, y)|x > 0} where (x1, Y1) O (x2, Y2) = (x1x2, Y1 + Y2) and c O (x, y) = (x°, cy) a vector space? O Yes O No
: Let V be the set of all ordered pairs of real numbers (u1, u). Consider the following addition and scalar multiplication operations on u = ("1, 42) and v = (v1, 2): %3D u +v = (41 + V1, U2+ v2), ku = - (41, ku2) Consider the following vector space axioms. (1) K(mu) = (km)u (ii) k(u + v) (iii) (k + m)u = ku + mu = ku + kv Which of the above axioms are true for the above set with the given operations? (A) (i) and (11) only (B) (1) only (C) none of them (D) (ii) and (ii) only (E) (1i) only (F) (i) only (G) all of them (H) (i) and (iii) only
Consider the set R²={(a,b)Da,b ER}. Suppose the following operations are defined on this set. Addition: (X 1.y 1)® (x 2Y2)=(x1+X2.Y1+Y2) • Scalar Multiplication: C © (x.y)=(cx,÷y) Which of the following axioms for a vector space fail to hold under these operations? O Distributivity of scalar multiplication over vector addition (c +d)O u=(c ©u)e (dO u) for all scalars C,d and all ordered pairs u=(x.y) 10 u=u for all u ER? O Closure under addition Additive identity QUESTION 15 Find the coordinates of X= (13, 26, 39) relative to the orthonormal basis B = inR 13 49 29 -78 O b.(x], =| 58 39 61 =| 58

SEE MORE QUESTIONS

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS