Prove Lemma
a) Fix a basis {v1, v2, . . . , vn} for an n-dimensional vector space V. Define a linear operator T : V → Fn in the following way: For each x = Σni=1 civi ∈ V, define
. Then T is a linear
operator.
b) Let T be a linear operator from V to W. Suppose that {v1, v2, . . . , vn} is a basis for V and {w1, w2, . . . , wm} is a basis for W. For each j with 1 ≤ j ≤ n, there exists a unique set of scalars a1j , a2j , . . . , amj such that Tvj = Σmi=1 aijwi . If we let A = [aij], then TA = T.
Homework Answers
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Prove Lemma
a) Fix a basis {v1, v2, . . . ,
vn} for an n-dimensional...
Let {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Ho...
Let {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Hom(V, W ) s.t. ∀ i ∈ {1, 2, ..., n}, α(vi) = wi
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the s...
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...
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors...
7. Let T : V → W be a linear transformation, and let v1,v2,...,vn be vectors in V. Suppose that T (v1), T (v2), . . . , T (vn) are linearly independent. Show that v1, v2, . . . , vn are linearly independent.
(4) Let {V1, V2, ..., Vn} be a basis for a vector space V. If w...
(4) Let {V1, V2, ..., Vn} be a basis for a vector space V. If w is an element of V whose coefficient vector is the zero vector, show that w must be the zero element.
Linear Aljebra Let B = {vy, V2, V3) be a basis for R in which we...
Linear Aljebra
Let B = {vy, V2, V3) be a basis for R in which we have and V3 Also, let TR-R be the linear operator such that: T(v.) = T(v2) and T(v.) = -0 X1 Part (a): Find a formula for T X₂ X, Answer: T X2 -0 [Ogg 912 943 = A x2 where A = 421 422 423 х3 231 232 233 Xz 0 } then find the following: Now let the vector w= Part (b): Find...
Let V be a vector space. Suppose dimV = n and {V1, V2, ..., Vn} is...
Let V be a vector space. Suppose dimV = n and {V1, V2, ..., Vn} is a basis of V. Thei which of the following is always true? a) Any set of n vectors is linearly dependent b) Any linearly dependent set in V is not part of basis c) Any linearly dependent set with n - 1 vectors is a basis d) A linearly independent set with n vectors is a basis
please help me with questions 1，2，3 1. Let V be a 2-dimensional vector space with basis...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
(1) Let S (v1, V2,..., Vn be a set of vectors in a vector space V....
(1) Let S (v1, V2,..., Vn be a set of vectors in a vector space V. (a) Define what does it mean for S to be linearly independent. (b) Define what does it mean for S to be a spanning set for V. (c) Define what does it mean for S to be a basis for v.
For given capacity of knapsack W and n items {i1,i2,...,in} with its own value {v1,v2,...,vn} and...
For given capacity of knapsack W and n items {i1,i2,...,in} with its own value {v1,v2,...,vn} and weight {w1,w2,...,wn}, find a greedy algorithm that solves fractional knapsack problem, and prove its correctness. And, if you naively use the greedy algorithm to solve 0-1 knapsack problem with no repetition, then the greedy algorithm does not ensure an optimal solution anymore. Give an example that a solution from the greedy algorithm is not an optimal solution for 0-1 knapsack problem.
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42...
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42 be the change- 01-1 of-coordinate matrix from V to W. Let A be the defining matrix for f under the basis W diagonalize A.
5. Given a linear map f R3R3 if V...
(4) Let {V1, V2, ..., Vn} be a basis for a vector space V. If w is an element of V whose coefficient vector is the zero vector, show that w must be the zero element.
Linear Aljebra
Let B = {vy, V2, V3) be a basis for R in which we have and V3 Also, let TR-R be the linear operator such that: T(v.) = T(v2) and T(v.) = -0 X1 Part (a): Find a formula for T X₂ X, Answer: T X2 -0 [Ogg 912 943 = A x2 where A = 421 422 423 х3 231 232 233 Xz 0 } then find the following: Now let the vector w= Part (b): Find...
Let V be a vector space. Suppose dimV = n and {V1, V2, ..., Vn} is a basis of V. Thei which of the following is always true? a) Any set of n vectors is linearly dependent b) Any linearly dependent set in V is not part of basis c) Any linearly dependent set with n - 1 vectors is a basis d) A linearly independent set with n vectors is a basis
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
(1) Let S (v1, V2,..., Vn be a set of vectors in a vector space V. (a) Define what does it mean for S to be linearly independent. (b) Define what does it mean for S to be a spanning set for V. (c) Define what does it mean for S to be a basis for v.
5. Given a linear map f R3R3 if V Vi, V2, va) is a basis of R3, and further, a) State the defining matrix of f under the basis vi, V2, vs) -3 2 0 b) Let W-(w1, w2, w3) be another basis of R3 and P42 be the change- 01-1 of-coordinate matrix from V to W. Let A be the defining matrix for f under the basis W diagonalize A.
5. Given a linear map f R3R3 if V...
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