Let X={v_1,v_2,v_3…,v_n} be a subset of a vector space V over F. Let A(X) denote the set of linear combinations of the form α_1 v_1+α_2 v_2+⋯+ α_n v_n, where α_1∈F and α_1+α_2…+ α_n=1. Prove that A(X) is a subspace of V if and only if v_i= 0_v for some i ∈{1,2,…,n}. Let X = {V₁, V₂, V3 ...,Vn} be a subset of a vector space V over F. Let A(X) denote the set of linearcombinations of the form a₁v₁ + a₂V₂ + ... + Anvn, where a₁ E F and a₁ + a₂...+ an = 1. Provethat A(X) is a subspace of V if and only if vi 0, for some i E {1,2,...,n}.1

Answered: Image /qna-images/answer/d0de553e-7a73-4e16-9aa1-b49187fe2101.jpg

Let X = {V₁, V₂, V3 ...,Vn} be a subset of a vector space V over F. Let A(X) denote the set of linear + an = 1. Prove combinations of the form a₁v₁ + a₂V₂ + ... + anvn, where a₁ E F and a₁ + a₂... that A(X) is a subspace of V if and only if v₁ = 0, for some i E {1,2,...,n}.

Let X = {V₁, V₂, V3 ...,Vn} be a subset of a vector space V over F. Let A(X) denote the set of linear + an = 1. Prove combinations of the form a₁v₁ + a₂V₂ + ... + anvn, where a₁ E F and a₁ + a₂... that A(X) is a subspace of V if and only if v₁ = 0, for some i E {1,2,...,n}.

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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Question

Let X={v_1,v_2,v_3…,v_n} be a subset of a vector space V over F. Let A(X) denote the set of linear combinations of the form α_1 v_1+α_2 v_2+⋯+ α_n v_n, where α_1∈F and α_1+α_2…+ α_n=1. Prove that A(X) is a subspace of V if and only if v_i= 0_v for some i ∈{1,2,…,n}.

Let X = {V₁, V₂, V3 ...,Vn} be a subset of a vector space V over F. Let A(X) denote the set of linear
combinations of the form a₁v₁ + a₂V₂ + ... + Anvn, where a₁ E F and a₁ + a₂...+ an = 1. Prove
that A(X) is a subspace of V if and only if vi 0, for some i E {1,2,...,n}.
1

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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