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Provide a proof for the following: Definition: A basis for subspace H of Rn is a linearly independent set of vectors in H that also spans H. Proof must show: (1) The zero vector is in H. (2) For each u, v ∈ H, the sum u + v ∈ H. (Closed under vector addition). (3) For each u∈H and each scalar c, the vector cu∈H. (Closed under scalar multiplication).

Provide a proof for the following: Definition: A basis for subspace H of Rn is a linearly independent set of vectors in H that also spans H. Proof must show: (1) The zero vector is in H. (2) For each u, v ∈ H, the sum u + v ∈ H. (Closed under vector addition). (3) For each u∈H and each scalar c, the vector cu∈H. (Closed under scalar multiplication).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.CR: Review Exercises
Problem 78CR: Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set...
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Provide a proof for the following:

Definition: A basis for subspace H of Rn is a linearly independent set of vectors in
H that also spans H.

Proof must show:

(1) The zero vector is in H.
(2) For each u, v ∈ H, the sum u + v ∈ H. (Closed under vector addition).
(3) For each u∈H and each scalar c, the vector cu∈H.
(Closed under scalar multiplication).
(4) The null space of an m × n matrix A is the set Nul A of all solutions to the homogeneous equation Ax = 0.

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