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6) Let S be a subset of an m-dimensional vector space and suppose S contains fewer...
3. Suppose S = {V1, V2, V3} is a linearly dependent subset of a vector space...
3. Suppose S = {V1, V2, V3} is a linearly dependent subset of a vector space V. Using only the definition of linear dependence and the span of a set, prove that you can remove one vector from S and still have a set with the same span of the original set.
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V...
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V of dimension seven. Explain why W1 n W3 {0 (b) Suppose V is a vector space of dimension 55, and let Wi and W2 be subspaces of V of dimension 36 and 28 respectively. What is the least possible value and the greatest possible value of dim(Wi + W2)?
Explain what a basis for a vector space is. How does a basis differ from a span of a vector space...
explain what a basis for a vector space is. How does a basis differ from a span of a vector space? What are some characteristics of a basis? Does a vector space have more than one basis? Be sure to do this: A basis B is a subset of the vector space V. The vectors in B are linearly independent and span V.(Most of you got this.) A spanning set S is a subset of V such that all vectors...
Problem 2. Let V be a five dimensional real vector space equipped with a non-degenerate symmetric...
Problem 2. Let V be a five dimensional real vector space equipped with a non-degenerate symmetric bilinear form (,). Suppose there exists a two dimensional subspace U of V such that (u, v) = 0 for all v EU. What are the possible signatures for (,)? Be sure to explain why each possible signature can or cannot occur.
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº =...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Prove that if a subset S of a vector space contains at least one nonzero vector, then spanS) cont...
Prove that if a subset S of a vector space contains at least one nonzero vector, then spanS) contains an infinite number of vectors. 29.
Prove that if a subset S of a vector space contains at least one nonzero vector, then spanS) contains an infinite number of vectors. 29.
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representati...
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
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)?...
If the number of vectors nin a vector space is less than the dimension m of...
If the number of vectors nin a vector space is less than the dimension m of the vector space they belong to, then: Select one or more: a. the set of vectors is always linearly dependent. b. the set of vectors can or cannot be linearly independent. c. the span of these vectors can or cannot span the vector space of dimension m. d. the set of vectors is always linearly independent e. the span of these vectors always span...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
3. Suppose S = {V1, V2, V3} is a linearly dependent subset of a vector space V. Using only the definition of linear dependence and the span of a set, prove that you can remove one vector from S and still have a set with the same span of the original set.
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V of dimension seven. Explain why W1 n W3 {0 (b) Suppose V is a vector space of dimension 55, and let Wi and W2 be subspaces of V of dimension 36 and 28 respectively. What is the least possible value and the greatest possible value of dim(Wi + W2)?
Problem 2. Let V be a five dimensional real vector space equipped with a non-degenerate symmetric bilinear form (,). Suppose there exists a two dimensional subspace U of V such that (u, v) = 0 for all v EU. What are the possible signatures for (,)? Be sure to explain why each possible signature can or cannot occur.
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Prove that if a subset S of a vector space contains at least one nonzero vector, then spanS) contains an infinite number of vectors. 29.
Prove that if a subset S of a vector space contains at least one nonzero vector, then spanS) contains an infinite number of vectors. 29.
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T : V -» V is matrix representation with respect to every basis of V. Prove that the dimension of linear transform ation that has the same that T must be a scalar multiple of the identity transformation. You can assume V is 3
Prob le m 5 (Bonus 2 points) Let V be a finite dimensional vector space. Suppose that T :...
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)?...
If the number of vectors nin a vector space is less than the dimension m of the vector space they belong to, then: Select one or more: a. the set of vectors is always linearly dependent. b. the set of vectors can or cannot be linearly independent. c. the span of these vectors can or cannot span the vector space of dimension m. d. the set of vectors is always linearly independent e. the span of these vectors always span...
15. Advanced problem: Let's say that a vector space X“splits” the spaces U and G if either Uç X & W or W ÇX V. a. Is there a vector space C that splits A = R^3 and B = {the x-axis in R^3} ? If there is, find it (no need to prove your claim) and if not, explain why it cannot exist. b. Suppose that U & W are a finite-dimensional. On what condition does there exist a...
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