Prove that if S={v1, v2, v3} is a linearly dependent set of vectors in a vector space V, and v4

step 1 In this problem, we need to prove that if the set S, which contains V1, V2, and V3, is a linearl

Prove that if S={v1, v2, v3} is a linearly dependent set of...

Prove that if $S=\left\{v_{1}, v_{2}, v_{3}\right\}$ is a linearly dependent set of vectors in a vector space $V,$ and $v_{4}$ is any vector in $V$ that is not in $S,$ then $\left\{v_{1}, v_{2}, v_{3}, v_{4}\right\}$ is also linearly dependent.

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Key Concepts

Linear Dependence

In the context of vector spaces, a set of vectors is said to be linearly dependent if there exists a nontrivial linear combination of these vectors that equals the zero vector. This means that at least one of the vectors in the set can be expressed as a combination of the others, demonstrating redundancy within the set.

Nontrivial Linear Combination

A nontrivial linear combination involves scalars, not all of which are zero, such that when they multiply their respective vectors and are summed, the result is the zero vector. The existence of such a combination is the defining criterion for linear dependence, distinguishing it from a trivial combination where all scalar multipliers are zero.

Extension by Additional Vectors

When extending a set of vectors by including an additional vector, any existing nontrivial linear dependence relation among the original vectors remains valid by assigning a zero coefficient to the new vector. This demonstrates that if a set is already linearly dependent, any superset that includes it is also linearly dependent, regardless of the properties of the additional vector.

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