True or false: If vectors $2u$, $3v$, and $4w$ are linearly independent, then $u$, $v$, and $w$, are also linearly independent. Explain.
This was a question on my test a second ago and I said false, because if $u$, $v$, or $w$ are the $0$ vector, then they are all linearly dependent by theorem $8$ in my textbook.
Because for constants $c_1, \dots, c_n$ if not all $c_i$'s are $0$, then
If $u$ is the zero vector then $c = 1$, then all other $c$ are $0$, and then the sum of vectors is $0$ --> linear dependent
did i do this right?