Re: Quick Matrix Algrebra Question
Think of A as the matrix and X as a single column vector.
ie , X={x1,x2,x3,...xn }T where T is the transpose .
And A can be written as {V1 V2 V3 ... Vn} and the Vi's are column vectors .
In other words , you wish to solve for x1*V1 + x2*V2 +...xn*Vn = ( 0,0,0)
The rhs is the 0 vector which may be written as (0,0,0) .
ie , V1=(0,0,1) V2= (1,0,1) V3= (0,1,1) and the xi's are scalars .
So we wish to solve for x1*(0,0,1) + x2*(1,0,1) + x3*(0,1,1) =(0,0,0)
Or x2=0 , x3=0 and x1+x2+x3=0 which implies x1=0 .
Since all three of the xi's are zero , this implies that the vectors V1,V2 and V3 are linearly independent .
Also it's important to know that a set of vectors is linear dependent if and only if some vi can be written as a linear combination of the others .If you can prove independence , then you can prove that a single vector cannot be written as a linear combination of the others .
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