Prove that if {}isany
linearly dependent set, then there is at least one
vectorxi in the set that can be expressed as a
linearcombination of the others and conversely.

See the proof below.

Step-by-step explanation:

What we need to proof is this: "Assuming X a vector space over a scalar field C. Let X= {x1,x2,....,xn} a set of vectors in X, where . If the set X is linearly dependent if and only if at least one of the vectors in X can be written as a linear combination of the other vectors"

Proof

Since we have a if and only if w need to proof the statement on the two possible ways.

If X is linearly dependent, then a vector is a linear combination

We suppose the set is linearly dependent, so then by definition we have scalars in C such that:

And not all the scalars are equal to 0.

Since at least one constant is non zero we can assume for example that , and we have this:

We can divide by c1 since we assume that and we have this:

And as we can see the vector can be written a a linear combination of the remaining vectors . We select v1 but we can select any vector and we get the same result.

If a vector is a linear combination, then X is linearly dependent

We assume on this case that X is a linear combination of the remaining vectors, as on the last part we can assume that we select and we have this:

For scalars defined in C. So then we have this:

So then we can conclude that the set X is linearly dependent.

And that complet the proof for this case.

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step-by-step explanation:

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2

step-by-step explanation: