(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

Question

Question

(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

(a) In the vector space, V = {f : R → R}, prove that the set {x⁹,sin5x,cos2x} is linearly independent.

(b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R³? Justify your answer.

math algebra

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

o) ton sidev the equafion e equain U) putting . 0 putting x=5 on Úi) 5 Sef Consider the e^uafon 3 Ci し2 0 I-2 det (A) 2o Multiplying the enuh on e- . the set 1,2/3 Indepen denl Set in i3 also the set Contains 3 elements which is elual to the dim&S/on Hence this Set il Ar

Add a comment

Answer 2

Similar Homework Help Questions

3. In the vector space V independent. R) prove that the set (cos 5, sin 3r,...

3. In the vector space V independent. R) prove that the set (cos 5, sin 3r, is linearly R
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...

Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1,...

1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1, 2+2-2,1 +32"} b. {2+1, 3x +3',-6 +2"} 8. Let T be a linear transformation from a vector space V to W over R. . Let .. . be linearly independent vectors of V. Prove that if T is one to one, prove that (un)....(...) are linearly independent. (m) is ) be a spanning set of V. Prove that it is onto, then Tu... h...

Problem 9. Let V be a vector space over a field F (a) The empty set...

Problem 9. Let V be a vector space over a field F (a) The empty set is a subset of V. Is a subspace of V? Is linearly dependent or independent? Prove your claims. (b) Prove that the set Z O is a subspace of V. Find a basis for Z and the dimension of Z (c) Prove that there is a unique linear map T: Z → Z. Find the matrix representing this linear map and the determinant of...
One of the following set of vectors are linearly independent Select one: O a. (1,2,3), (0,1,0),(0,0,1),(1,...

One of the following set of vectors are linearly independent Select one: O a. (1,2,3), (0,1,0),(0,0,1),(1, 1, 1) O b. x, 1,x2 +1. (1, 1, 2, 1.4). (2.-1.2,-1,6), (0.0.0.0.0) d. (1.1.2.1.4). (2.2. 4.2.8) For any finite n-dimensional vector space V with a basis B Select one: a. A subspace of V is a subset of V that contains a zero vector and is closed under the operation of addition b. None C. The coordinate vector of any vector v in...
For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space...

For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space and {11,...,Vk} is a linearly independent set in V, then k sn. C5 Every basis for P2(R) has exactly two vectors in it. C6 If {V1, V2} is a basis for a 2-dimensional vector space V, then {ağı + bū2, cũı + dv2} is also a basis for V for any non-zero real numbers a,b,c,d.

3. Suppose S is a linearly independent generating set for a vector space V . Show...

3. Suppose S is a linearly independent generating set for a vector space V . Show that S is an efficient generating set, i.e., any proper subset of S is not a generating set.
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for...

Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal co...

2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal complement of the space 0 Span 2,2 Do not simply compute the cross product. (c) Let A be a 5 × 2 rnatrix with linearly independent columns. Using the rank-nullity theorem applied to AT, and any other results from the course, find the dinension of Col(A) 2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector...

suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).

suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).

(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

Homework Answers

Add Answer to:
(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

Post as a guest

Earn Coins

3. In the vector space V independent. R) prove that the set (cos 5, sin 3r,...

Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...

1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1,...

Problem 9. Let V be a vector space over a field F (a) The empty set...

One of the following set of vectors are linearly independent Select one: O a. (1,2,3), (0,1,0),(0,0,1),(1,...

For Problems C4-C11, prove or disprove the statement. C4 If V is an n-dimensional vector space...

3. Suppose S is a linearly independent generating set for a vector space V . Show...

Problem 4. Let n E N. We consider the vector space R” (a) Prove that for...

2. (a) Show that is an orthogonal basis for R3. (b) Find a non-zero vector v in the orthogonal co...

suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).

(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

Homework Answers

Add Answer to: (a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.

Post as a guest

Earn Coins

Add Answer to:
(a) In the vector space, V = {f : R → R}, prove that the set {x9,sin5x,cos2x} is linearly independent. (b) Is {(1,2,3),(−2,1,0),(1,0,1)} a basis for R3? Justify your answer.