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2. For each space below, give an example of a set that does not span the...
(a) This exercise will give an example of a connected space which is not locally connected. In th...
(a) This exercise will give an example of a connected space which is not locally connected. In the plane R2 , let X0 = [0, 1] × {0}, Y0 = {0} × [0, 1], and for each n ∈ N, let Yn = {1/n} × [0,1]. Let Y = X0 ∪ (S∞ n=0 Yn). as a subspace of R 2 with its usual topology. Prove that Y is connected but not locally connected. (Note that this example also shows that...
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the...
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector...
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector space" and "subspace" as precisely as you can. (b) Consider a line through the origin in R2, for example, the r-axis. Explain why this line is, or is not, a subspace of R2 in terms of your definitions in (a). (c) Consider the union of two lines through the origin in R2, for example, the z- and y-axes. Explain why this union of lines...
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 sta...
Hi,
could you post solutions to the following questions. Thanks.
2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
?24) 1. To show that set of vectors of formm 2. (a) if A is invertible,...
?24) 1. To show that set of vectors of formm 2. (a) if A is invertible, list three different methods to solve equation Ax-b. b) Application each of above mentioned methods to solve is a subspace of a space of all 2x2 matrices. -x2 +2x3 =0 Find a matrik that reflects vectors in R' about yz-plane and then expand the length twice (2 0 3 2 7 Given set of vectors S=(1 1, 1-1, 1 .. D in R 10)...
(5 pts) Give an example of a relation on a set that is a) both symmetric...
(5 pts) Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric. (3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n =1 An = {0} (b) Um_1 An = [0, 1] (c) n =1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either provide a proof showing the set is a subspace of R3, or provide a counterexample showing it is not a subspace: [9 marks] (i) S= {(x, y, z) ER3 : 4.0 + 9y + 8z = 0} (ii) S = {(x, y, z) E R3 : xy = 0} (b) Determine for which values of b ER, the set S = {(x, y, z)...
Problem 7.7. Give an example of a space that is connected, but not path con- nected. Problem 7.8....
want proof for theorem 7.12 using definition 7.9
Problem 7.7. Give an example of a space that is connected, but not path con- nected. Problem 7.8. Show that R" is not homeomorphic to R if n>1 Definition 7.9. Let be a point in X. Then X is called locally path connected at a if for each open set U containing r, there is a path connected open set V containing r such that V CU. If X is locally path...
Wite **the sum of two vectons, one in Span {u) and one in Span (wa). Assume...
Wite **the sum of two vectons, one in Span {u) and one in Span (wa). Assume that (.....) is an orthogonal besis Type an integer or simplified traction for each max element) Verity that {.uz) is an orthogonal sot, and then find the orthogonal projection of y onto Span(uz) y To verty that (0-uz) as an orthogonal set, find u, uz 2-0 (Simplify your answer.) The projection of yonte Span (0,2) 0 (Simplify your answers.) LetW be the subspace spanned...
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector space" and "subspace" as precisely as you can. (b) Consider a line through the origin in R2, for example, the r-axis. Explain why this line is, or is not, a subspace of R2 in terms of your definitions in (a). (c) Consider the union of two lines through the origin in R2, for example, the z- and y-axes. Explain why this union of lines...
Hi,
could you post solutions to the following questions. Thanks.
2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
?24) 1. To show that set of vectors of formm 2. (a) if A is invertible, list three different methods to solve equation Ax-b. b) Application each of above mentioned methods to solve is a subspace of a space of all 2x2 matrices. -x2 +2x3 =0 Find a matrik that reflects vectors in R' about yz-plane and then expand the length twice (2 0 3 2 7 Given set of vectors S=(1 1, 1-1, 1 .. D in R 10)...
(5 pts) Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric. (3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n =1 An = {0} (b) Um_1 An = [0, 1] (c) n =1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category...
QUESTION 2. (a) Decide whether each of the following subsets of R’ is a subspace. Either provide a proof showing the set is a subspace of R3, or provide a counterexample showing it is not a subspace: [9 marks] (i) S= {(x, y, z) ER3 : 4.0 + 9y + 8z = 0} (ii) S = {(x, y, z) E R3 : xy = 0} (b) Determine for which values of b ER, the set S = {(x, y, z)...
want proof for theorem 7.12 using definition 7.9
Problem 7.7. Give an example of a space that is connected, but not path con- nected. Problem 7.8. Show that R" is not homeomorphic to R if n>1 Definition 7.9. Let be a point in X. Then X is called locally path connected at a if for each open set U containing r, there is a path connected open set V containing r such that V CU. If X is locally path...
Wite **the sum of two vectons, one in Span {u) and one in Span (wa). Assume that (.....) is an orthogonal besis Type an integer or simplified traction for each max element) Verity that {.uz) is an orthogonal sot, and then find the orthogonal projection of y onto Span(uz) y To verty that (0-uz) as an orthogonal set, find u, uz 2-0 (Simplify your answer.) The projection of yonte Span (0,2) 0 (Simplify your answers.) LetW be the subspace spanned...
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