Question

In: Advanced Math

Let S be a subset of a vector space V . Show that span(S) = span(span(S))....

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.

Solutions

Expert Solution


Related Solutions

Let U be a subset of a vector space V. Show that spanU is the intersection...
Let U be a subset of a vector space V. Show that spanU is the intersection of all the subspaces of V that contain U. What does this say if U=∅? Need proof
Proof: Let S ⊆ V be a subset of a vector space V over F. We...
Proof: Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.
Let T and S be linear transformations of a vector space V, and TS=ST (a) Show...
Let T and S be linear transformations of a vector space V, and TS=ST (a) Show that T preserves the generalized eigenspace and eigenspace of S. (b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)2 (t-3)2?. What is the jordan canonical form of S. (c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.
(10pt) Let V and W be a vector space over R. Show that V × W...
(10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W and λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv. (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)
If V is a linear space and S is a proper subset of V, and we...
If V is a linear space and S is a proper subset of V, and we define a relation on V via v1 ~ v2 iff v1 - v2 are in S, a subspace of V. We are given ~ is an equivalence relation, show that the set of equivalence classes, V/S, is a vector space as well, where the typical element of V/S is v + s, where v is any element of V.
Let V be a vector space and let U and W be subspaces of V ....
Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .
Let V be a finite dimensional vector space over R. If S is a set of...
Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?
1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S...
1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S be in- vertible. Prove λ ∈ C is an eigenvalue of T if and only if λ is an eigenvalue of STS−1. Give a description of the set of eigenvectors of STS−1 associated to an eigenvalue λ in terms of the eigenvectors of T associated to λ. Show that there exist square matrices A, B that have the same eigenvalues, but aren’t similar. Hint:...
Let A be a closed subset of a T4 topological space. Show that A with the...
Let A be a closed subset of a T4 topological space. Show that A with the relative topology is a normal T1
6. Let V be the vector space above. Consider the maps T : V → V...
6. Let V be the vector space above. Consider the maps T : V → V And S : V → V defined by T(a1,a2,a3,...) = (a2,a3,a4,...) and S(a1,a2,a3,...) = (0,a1,a2,...). (a) [optional] Show that T and S are linear. (b) Show that T is surjective but not injective. (c) Show that S is injective but not surjective. (d) Show that V = im(T) + ker(T) but im(T) ∩ ker(T) ̸= {0}. (e) Show that im(S) ∩ ker(S) = {0}...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT