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No. $$X_1=(1,0,0)$$ $$X_2=(0,1,0)$$ $$Y_1=(-1,0,0)$$ $$Y_2=(0,-1,0)$$ $$V=(0,0,1)$$ [Plot via Wolfram Alpha] Edit for the reposed question: Answer to the first part is still no. $$X_1=(1,0,0,0,0)$$ $$X_2=(0,1,0,0,0)$$ $$Y_1=(0,0,1,0,0)$$ $$Y_2=(0,0,0,1,0)$$ $$V=(0,0,0,0,1)$$ All vectors are linearly independent (and orthogonal). As for $Y_3$, yes, we ...


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The vector, v= <0, 0, 1> is perpendicular to X1= <1, 0, 0> and X2= <0, 1, 0> as well as to Y1= <1, 1, 0> and Y2= <1, -1, 0> but those four vectors are not independent. I don't know what you mean by "uncorrelated" vectors.



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