Prove that if S={v1, v2, …, vr} is a linearly dependent set of vectors in a vector space V, and

step 1 In this problem, we need to show that if the set S, which has v1, v2, up to vr, is a linearly de

Prove that if S={v1, v2, …, vr} is a linearly dependent set...

Prove that if $S=\left\{v_{1}, v_{2}, \ldots, v_{r}\right\}$ is a linearly dependent set of vectors in a vector space $V,$ and if $v_{r+1}, \ldots, v_{n}$ are any vectors in $V$ that are not in $S,$ then $\left\{v_{1}, v_{2}, \ldots, v_{r}, v_{r+1}, \ldots, v_{n}\right\}$ is also linearly dependent.

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Key Concepts

Nontrivial Solution

A nontrivial solution to a homogeneous linear equation is one in which the scalars (coefficients) are not all zero. The existence of such a solution when forming a linear combination of vectors that results in the zero vector is the defining characteristic of linear dependence.

Superset Property of Linear Dependence

The superset property of linear dependence states that if a set of vectors is linearly dependent, then any larger set that contains this set remains linearly dependent. This is because the dependent relationship found in the smaller set is preserved when additional vectors are added, regardless of whether those new vectors contribute to the dependence.

Linear Dependence

Linear dependence is a property of a set of vectors in which there exists a nontrivial linear combination of the vectors that equals the zero vector. This means at least one vector in the set can be expressed as a sum of multiples of the other vectors, indicating a redundancy among the vectors in the space.

Linear Combination

A linear combination involves multiplying vectors by scalars and then adding the results together. This concept is fundamental in linear algebra because it allows us to express vectors in terms of others, assess relationships among them, and formalize the ideas of span, basis, and dependence within a vector space.

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