Question

give example of two linearly independent sets in vector space is linearly independent ​

Answer 1

Answer:

Hey guys

Step-by-step explanation:

Let U and V be two subspaces of a vector space E such that U∩V={0}. Prove that if X⊆U and Y⊆V are two linearly independent sets then so is X∪Y.

So I believe I have a good start, but it may not hold (I'm still a beginner at conducting proofs..). Feel free to critique wherever necessary!

Origonal Proof:

X and Y are linearly independent. Thus, they do not contain the 0 element. X is a subset of U and Y is a subset of V, and V∩U={0}, so X∩Y={∅}. (So, for any x∈X∪Y, x≠0)

Suppose X∪Y is linearly dependent.

So, there exists x∈X∪Y and y∈X∪Y such that x=y and they exist in the linear combination of other elements ∈X∪Y.

i.e. x=(λa+μb)=y for a,b∈X∪Y and scalers λ and μ in the Field.

If (λa+μb)=0, then we are done, for x≠0≠y.

Let (λa+μb)≠0. (Note: a≠0≠b, and λ≠0≠μ)

Then a and b are in a linearly independent set, so X∪Y is linearly independent

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Answer 2

Answer:

We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant. The set is of course dependent if the determinant is zero....

Step-by-step explanation:

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