Problem 5. Let V be a vector space and W₁, W₂ be two subspaces of V. DefineW₁ + W₂ = {x + y: x W₁ and y E W₂}.(a) Prove that W₁ + W₂ is a subspace of V that contains both W₁ and W₂.(b) If W₁, W₂ are contained in the subspace W of V, then W₁ + W₂ is also contained in W.1 Problem 4. Prove that l¹(N)RN.= {(x0, x1,..).(x0, x₁, ...) € RN : Σ\x₂] <∞E1Σ|xx | < 00 is a vector subspace ofn=0

Answered: Image /qna-images/answer/e2c1fe01-d601-417b-89d7-98f41a0be992.jpg

Answered: Problem 5. Let V be a vector space and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let W be the subspace spanned by u₁ and u₂, and write y as the sum of a vector in W and a vector orthogonal to W. -0- y = -2 3,4₁ 5 The sum is y = y + z, where y= (Simplify your answers.) -3 -1 is in W and Z= is orthogonal to W.
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. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v EV can be represented uniquely as v = v₁ + v2 where V₁ € W₁, V2 € W2. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V = W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that B₁ U B2 is a basis of V.
solve parts c and d
1. State if the following statement is true or false. The set of vectors B = {u= (1,1,4) ;v = (2,0,2);w = (0,2,1)} is linearly independent in the vector space R³. O True O False
4. Let V be a vector space. (a) If vo E V, is the subset W := {kvo|k e R} a subspace of V? Justify your answer. (b) If x e V and y e V, is the subset U := {sx + ty s, t e R} a subspace of V? Justify your answer.
V = {(x, y) E R² | x = y} u {(x, y) E R² | x = = -2y}. Complete the following statements to determine if V is a subspace of R. (a) V is ? If it is non-empty, give two different examples of vectors in V. If it is empty, then leave the following spaces blank. example 1: aа — example 2: b = Note: Normally, only one example is required to show V is not empty in a proof. (b) V is ? v under vector addition. If it is not closed, enter two vectors a, b E V below, whose sum is not in V. If it is closed, then leave the following spaces blank. a = b = (c) V is ? under scalar multiplication. If it is not closed, enter a scalar k and a vector c E V below, whose product is not in V. If it is closed, then leave the following spaces blank. (OO) k = and c = (d) V ? of R?.
n=U Problem 5. Let V be a vector space and W₁, W₂ be two subspaces of V. Define W₁ + W₂ = {x + y: x e W₁ and ye W₂}. (a) Prove that W₁ + W₂ is a subspace of V that contains both W₁ and W₂. (b) If W₁, W₂ are contained in the subspace W of V, then W₁ + W₂ is also contained in W. Problem 6. Let W = {(a1, a2, a3) € R³: a₁ = 3a2 and a3 = -a2}. Is W a subspace of R³ under the operations of addition and scalar multiplication defined on R³? Justify your answers.
4. Let V and W be finite dimensional vector spaces. Let T : V –→ W be a linear transformation of V into W. (a) Let H be a subspace of V. Let T(H) be the set of all images of the vectors in H under the transformation T. Show that T(H) is a subspace of W and dim T(H) < dim H (b) Now suppose that T is one-to-one. Show that dim T(H)= dim H.
Determine if the given Vector Subspace is LINEARLY INDEPENDENT IN THE REQUESTED VECTOR SPACE: [1 2] [5 (Item 24) }|0 1 41 [3 0] 5 ,1 3 1 in (M(3x2) [R], +;). 1 4) [4 10] l2 3. (Item 26){ sin (2x), cos(2x), sin (x) · cos(x) } in (F[0,T], R|], +;). note: (F[1, R], +,;) Vector Space of Continuous Functions over Reals : ("R"). Vector Space of Matrices of "m rows" and "n columns" over the Reais ("R"). (M[R], +,;) (m xn)
Which of the listed pairs has the set U as a vector subspace of the vector space V?

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