1- Let W1, W2 be two subspaces of a vector space V . Show that both...

1- Let W1, W2 be two subspaces of a vector space V . Show that
both W1 ∩ W2 and W1 +W2 are subspaces.?and Show that W1 ∪ W2 is a subspace
only when W1 ⊂ W2 or W2 ⊂ W1.
(recall that W1 + W2 = {x + y | x ∈ W1, y ∈ W2}.)

Expert Solution

Colby Messinger answered 5 months ago

Let V be a vector space, and suppose that U and W are both subspaces of...

Let V be a vector space, and suppose that U and W are both subspaces of V. Show that U ∩W := {v | v ∈ U and v ∈ W} is a subspace 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 .

Suppose that W1 and W2 are subspaces of V and dimW1<dimW2. Prove that there is a...

Suppose that W1 and W2 are subspaces of V and dimW1<dimW2. Prove that there is a nonzero vector in W2 which is orthogonal to all vectors in W1.

Suppose that ? and ? are subspaces of a vector space ? with ? = ?...

Suppose that ? and ? are subspaces of a vector space ? with ? = ? ⊕ ?. Suppose also that ??, … , ?? is a basis of ? amd ??, … , ?? is a basis of ?. Prove ??, … , ??, ??, … , ?? is a basis of V.

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

(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.)

QUESTION 1 Vector Space Axioms Let V be a set on which two operations, called vector...

QUESTION 1 Vector Space Axioms Let V be a set on which two operations, called vector addition and vector scalar multiplication, have been defined. If u and v are in V , the sum of u and v is denoted by u + v , and if k is a scalar, the scalar multiple of u is denoted by ku . If the following axioms satisfied for all u , v and w in V and for all scalars k...

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.

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.

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}...

Question