Prove Linear Dependence for Set of Vectors w/ Zero Vector

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SUMMARY

Any set of vectors that includes the zero vector is linearly dependent. This is proven by considering the equation (C1)(X1) + (C2)(X2) + ... + (Cn)(Xn) = 0, where X1 is the zero vector. In this scenario, C1 can take any value while all other constants (C2 to Cn) are zero, demonstrating that not all coefficients need to be zero for the equation to hold true. Thus, the presence of the zero vector guarantees linear dependence.

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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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