Prove Linear Dependence for Set of Vectors w/ Zero Vector

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Any set of vectors that includes the zero vector is linearly dependent because the zero vector can be expressed as a linear combination of the vectors with non-zero coefficients. Specifically, if the zero vector is included, one can set its coefficient to any non-zero value while keeping the coefficients of the other vectors as zero, leading to a valid linear combination that equals zero. This demonstrates that not all coefficients need to be zero for the set to be dependent. The explanation aligns with standard linear algebra principles, confirming the dependency of such sets. Understanding this concept is crucial for grasping linear algebra fundamentals.
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I have to prove that any set of vectors containing the zero vector is linearly dependent.
How can I approach this?
 
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(C1)(X1) + (C2)(X2) + ... + (Cn)(Xn) = 0

Take X1 = 0 vector; and all other constants besides C1 to be zero.

(C1)(0) + (0)(X2) + ... +(0)(Xn) = 0

In this case, C1 does not have to be equal to 0, it could be any number. In order for them to be independent, c1=c2=cn=0; since c1 does not equal 0 therefore they're dependent.

This is the way my professor described it to me, hopefully this helps.
 
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thats it? thanks a lot bro, I get it now. \m/
 

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