Prove Linear Dependence for Set of Vectors w/ Zero Vector

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In summary, to prove that any set of vectors containing the zero vector is linearly dependent, one approach is to assign X1 as the zero vector and set all other constants to zero. This results in a linear combination where C1 can be any number, but in order for the vectors to be independent, all C values must equal zero. This explanation was given by the professor.
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eyehategod
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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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  • #3
thats it? thanks a lot bro, I get it now. \m/
 

FAQ: Prove Linear Dependence for Set of Vectors w/ Zero Vector

What does it mean for a set of vectors to be linearly dependent?

Linear dependence means that one or more vectors in the set can be written as a linear combination of the other vectors in the set. In other words, one or more vectors in the set are redundant and do not add any new information.

Why is it important to prove linear dependence for a set of vectors?

Proving linear dependence allows us to determine if a set of vectors is a basis for a vector space. It also helps us identify if there are any redundant vectors in the set.

What is the role of the zero vector in proving linear dependence?

The zero vector is important because it is always linearly dependent with any set of vectors. If the zero vector is in the set, it automatically makes the set linearly dependent.

How can we prove linear dependence for a set of vectors?

We can prove linear dependence by setting up a system of equations and solving for the coefficients of the linear combination of the vectors. If there are infinitely many solutions or if one of the coefficients is zero, then the set is linearly dependent.

Can a set of vectors be linearly dependent without the zero vector?

Yes, a set of vectors can be linearly dependent without the zero vector. This means that there is a linear combination of the vectors that equals the zero vector without all of the coefficients being zero.

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