Linear Algebra: Let x1, x2, and x3 be linearly independent vectors in R^n, and let

y1 = x1+x2, y2 = x2+x3, y3 = x3+x1

Are y1, y2, y3 linearly independent? Prove your answer.

Question

Linear Algebra: Let x1, x2, and x3 be linearly independent vectors in R^n, and let

y1 = x1+x2, y2 = x2+x3, y3 = x3+x1

Are y1, y2, y3 linearly independent? Prove your answer.

agreene · Answer

I dont think any of them can be dependent because (If I'm remembering correctly) to be dependent, they have to be a linear combination, that is if y1 were y1= 2x1+x2 then it would be dependent. so, since the vectors are by definition linearly independent and the equations do not multiply by scalars they are all independent. (again, this is if I remember the definition accurately).