Suppose that the vectors in the set $[(x_1,x_2,x_3),(y_1,y_2,y_3),(z_1,z_2,z_3)]$ where each vector belongs to $R^3$ are linearly independent. Show that the vectors in the set $[(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)]$ are linearly independent. Can this problem be generalized for $R^n$ where $n$ is any positive integer? Please do not use any concept of matrices here, only use properties of vector spaces in general.
I may want to add that I tried but the correct mathematical procedure is not hitting me.