Problem 13 (10 pts) If L :V + W is a linear transformation of vector spaces...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
For the following six questions, indicate whether the following statements are true or false. In each case give a reason for your answer. Problem 13 [10 pts) If L:V +W is a linear transformation of vector spaces and U CW is a subspace of W, then {v € V | L(v € U} CW is a subspace of V. Problem 14 (10 pts) The set {A E R2x2 | A is nonsingular} is a subspace of R2x2. Problem 15 (10...
Problem 5: Let V and W be vector spaces and let B = {V1, V2, ..., Un} CV be a basis for V. Let L :V + W be a linear transformation, and let Ker L = {2 € V: L(x)=0}. (a) If Ker L = {0}, show that C = {L(v1), L(02), ..., L(vn) } CW is a linearly independent set in W. (b) If C = {L(01), L(V2),..., L(Un)} C W is a linearly independent set in W,...
THEOREM 3.4. Suppose T: V -» W is a linear transformation from K-linear spaces V to W. Then (a) ker(T) is a subspace of V, and (b) im(T) is a subspace of W PROOF. The proof is left as an exercise.
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Let V and W be finite dimensional vector spaces over R and T:V + W be linear. Let V be a subspace of V and Wo = T(V). (Select ALL that are TRUE) If T is surjective then Vo = {v EV : there is w E Wo such that T(v) = w} If T is injective then dim(VO) = dim(W). dim(ker(T) n Vo) = dim(VO) - dim(Wo).
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...
Problem 3 (Inner Products). (a) Let V, W be two finite dimensional vector spaces, dim V = n, dim W-m and V x W-+ R be a bilinear function, i.e., for each a V and b E W: 1(a, r-Ay)-I(a,r) + λ|(a, y), for all r, y W, λ ε R and 1(u + λν, b)-1(u, b) + λ|(u, b), for all u, u ε ν, λ ε R. Thus for each fixed a E V, W 14-R is a...
W is a rele that A linear transformation T from a vector space V into a vector space assigns to each vector 2 in V a unique vector T() in W. such that (1) Tutu = Tu+Tv for all uv in V, and (2) Tſcu)=cT(u) for all u in V and all scalar c. *** The kernel of T = {UE V , T(U)=0} The range of T = {T(U) EW , ue V } Define T :P, - R...
Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...