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- Homework Statement
- Let ##\{u_1,\ldots,u_s\}##and ##\{v_1,\ldots,v_t\}## be two sets of vectors. If ##s>t## and ##u_i## is a linear combination of ##v_1,\ldots, v_t##, show ##u_1,\ldots,u_s## is linearly dependent

- Relevant Equations
- Linear combination: c1u1+c2u2+c3u3+...+cnun=0

Linearly dependent is when there is a nontrival solution of the linear combination.

Hi Everybody,

I am having some difficulties on the prove this problem.

I picked a nice example when I was trying to think about the proof.

Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear combination of u in terms of v:

##K1(c1v1+c2v2)+K2(c3v1+c4v2)+K3(c5v1+c6v2)=0## It implies this system of linear equations for c1,..c6:

(K1c1+c3K2+c5K3)=0

(K1c2+c4K2+K3c6)=0

Here is where I am lost. What should I do next? Solve for K1,K2,K3? or something else.

Will this example help me prove this exercise?

Thanks,

cbarker 1

I am having some difficulties on the prove this problem.

I picked a nice example when I was trying to think about the proof.

Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear combination of u in terms of v:

##K1(c1v1+c2v2)+K2(c3v1+c4v2)+K3(c5v1+c6v2)=0## It implies this system of linear equations for c1,..c6:

(K1c1+c3K2+c5K3)=0

(K1c2+c4K2+K3c6)=0

Here is where I am lost. What should I do next? Solve for K1,K2,K3? or something else.

Will this example help me prove this exercise?

Thanks,

cbarker 1