I know what linear independence is and that if the solution set is non-zero, then the system is not linearly independent. But I am a bit confused with this particular question,
If I have 3 vectors, $X_1, X_2$ and $X_3$ that are linearly independent, and I have 3 more, $Y_1= X_1+X_2, Y_2= 2X_2-X_3$, and $Y_3= X_1+X_2-2X_3$, how do I show that the $Y$ vectors are also linearly independent? I know that the determinant has to be non-zero in order for the system to be linearly independent, but I don't think you have to find the determinant here? I am a bit confused. If anyone would help, it would be really appreciated. Thank you.