If W ⊆ V is a subspace of a vector space V over F, then for any v1, v2 ∈ V , we say v1 ∼ v2 if and only if v1 − v2 ∈ W. Check that ∼ is an equivalence relation on V .
Question 1. Let V be a finite dimensional vector space over a field F and let W be a subspace of Prove that the quotient space V/W is finite dimensional and dimr(V/IV) = dimF(V) _ dimF(W). Hint l. Start with a basis A = {wi, . . . , w,n} for W and extend it to a basis B = {wi , . . . , wm, V1 , . . . , va) for V. Hint 2. Our goal...
Let W be a subspace of an n-dimensional vector space V over C, and let T:V V be a linear transformation. Prove that W is invariant under T if and only if W is invariant under T- I for any i EC.
Problem #6. Let V be a finite dimensional vector space over a field F. Let W be a subspace of V. Define A(W) e Vw)Vw E W). Prove that A(W) is a subspace of (V).
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
2. Let V and W be vector spaces over F. Define the set v, w) |v V andwEW This is called the product of V and W (a) Show that V x W is a vector space. (b) Define a map w : V → V × W by w (z) = (z,0). Show that w is an injective linear map. Note that we can define a similar map lw (c) If (d) Show that V x W. (V W...
(3) Let V denote a vector space over the field F and let v,..., Un E V. (a) Show that span(vn, 2,. , Un) (b) Show that span (ui , U2 , . . . , vn) span(v)+ +span(vn). span(v1)@span(v2)㊥·..㊥8pan(vn) if and only if (vi , . , . , %) is linearly independent.
DETAILS LARLINALG8 4.R.023. Determine whether W is a subspace of the vector space V. (Select all that apply.) W = {f: f(0) = -1}, V = C[-1, 1] W is a subspace of V. W is not a subspace of V because it is not closed under addition. W is not a subspace of V because it is not closed under scalar multiplication.
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
solution to 2
(ii) Show that the image of f is not a subspace of R 2. Let U, V, and W be vector spaces over the field k, and let f: Ux V- W be a bilinear map. Show that the image of f is a union of subspaces of W. 3. Let k be a field, and let U, V, and W be vector spaces over k. Recall that
(ii) Show that the image of f is not...
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Problem 4. Let V be a vector space and let U and W be two subspaces of V. Let (1) Prove that ifU W andWgU then UUW is not a subspace of V (2) Give an example of V, U and W such that U W andWgU. Explicitly verify the implication of the statement in part1). (3) Proue that UUW is a subspace of V if and only if U-W or W- (4) Give an example that proues the...