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Chapter 4: Problem 59

Prove that any set of vectors containing the zero vector is linearly dependent.

Short Answer

Expert verified
Any set of vectors containing the zero vector is linearly dependent. The contradiction in assuming the set to be linear independent proves this.

Step by step solution

01

Define the given

A set of vectors \( \{v_1, v_2, \dots, v_n, 0\} \) is given, where \( 0 \) is the zero vector and \( n \) is any natural number.
02

Apply definition of linear dependence

By definition, vectors are linearly dependent if one vector can be expressed as a scalar multiple of another. Here, we can express the zero vector as a scalar multiple of any vector in the set. \(0 = \lambda \times v_i \) for any \( \lambda \) in real numbers and any \( i \) from 1 to \( n \).
03

Proof through contradiction

Let's suppose for the sake of contradiction that the vector set is linearly independent. Then, no non-trivial linear combination of the vectors can give the zero vector. However, if we take the scalar \( \lambda = 0 \), we can get the zero vector, \( 0 = 0 \times v_i \), which contradicts our assumption. Thus, the set is linearly dependent

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