(a) Prove that if 𝐯1, …, 𝐯m are linearly independent, then every subset, e.g., 𝐯1, …, 𝐯k with k<

step 1 For part A, suppose S contained in V is linearly dependent and T contains S. We're going to prov

(a) Prove that if 𝐯1, …, 𝐯m are linearly independent, then...

(a) Prove that if $\mathbf{v}_1, \ldots, \mathbf{v}_m$ are linearly independent, then every subset, e.g., $\mathbf{v}_1, \ldots, \mathbf{v}_k$ with $k<m$, is also linearly independent. (b) Does the same hold true for linearly dependent vectors? Prove or give a counterexample.

Key Concepts

Linear Dependence

A set of vectors is said to be linearly dependent when at least one vector in the set can be written as a linear combination of the others. This concept indicates that there are redundant vectors in the set, meaning that the vectors do not all contribute uniquely to spanning the vector space.

Linear Independence

Linear independence is a fundamental concept in linear algebra which states that a set of vectors is linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. This property implies that the only solution to the corresponding homogeneous linear equation is the trivial solution where all coefficients are zero.

Subsets of Linearly Independent Vectors

If a set of vectors is linearly independent, then any subset of that set is also linearly independent. This is because removing vectors from a linearly independent set cannot introduce a dependency relationship that was not already present, since any non-trivial linear combination yielding zero would contradict the independence of the original larger set.

Counterexamples in Linear Dependence

When dealing with linearly dependent vectors, taking a subset does not necessarily result in a linearly dependent set. It is possible to remove the redundant vectors in such a way that the remaining subset is linearly independent. Constructing such counterexamples is a common method in mathematical proofs and problem solving to demonstrate that properties of the whole set do not necessarily transfer to every subset.

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