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 {v1, v2, ..., vn} and {w1, w2, ..., wn} be bases of V and W , respectively. Prove: ∃ ! α ∈ Ho...
Let B = [V1, V2, V3] and B' = [W1, W2, W3] be bases for a vector space V and Vi = W1 + 5W2 – W3 U2 = W1 U3 -W1 - 4w2 – 2w3 If (U)b = (1,-1,2), then the coordinates of v relative to the basis B' are c1 = C2 = and cz
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, . . . ,...
UCICI 26. Let S = {V1, V2} and T = {w1, W2} be ordered bases for R?, where 38. If the transition matrix from S to T is determine T.
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.
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.
P4. Prove: If V = {V1, V2, ...,V) is a linearly indepen- dent set of vectors in R", and if W = {Wx+1, ...,wn} is a basis for the null space of the matrix A that has the vectors V1, V2, ..., Vk as its successive rows, then VUW = {V1, V2, ..., Vk, Wk+1,...,w.} is a basis for R". [Hint: Since V UW contains n vectors, it suffices to show that VUW is linearly independent. As a first step,...
(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.
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...
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 {v1, v2,v3} be a linearly independent set in R^n and let v = -αv3 +v1,w = v2 - αv1, u= v3-αv2 where αER, find all the values of α, where v, w, u are linearly dependent. do not use matrices.