1. ## Linearly Independent Vectors

Problem:
Consider the vectors $\displaystyle \vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}$ in $\displaystyle \mathbb{R}^{n}}$, with $\displaystyle \vec{v_{m}} = \vec{0}$. Are these vectors linearly independent?

My solution:
No, they are not. Since $\displaystyle \vec{v_{m}} = \vec{0}$, $\displaystyle rank[\vec{v_{1}} ... \vec{v_{m}}]$ will fail to equal m. There will not be a pivot in every column, which is a requirement for a set of linearly independent vectors.

Am I on the right track? Any input appreciated!

2. Originally Posted by tangibleLime
Consider the vectors $\displaystyle \vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}$ in $\displaystyle \mathbb{R}^{n}}$, with $\displaystyle \vec{v_{m}} = \vec{0}$. Are these vectors linearly independent?
No, they are not. Since $\displaystyle \vec{v_{m}} = \vec{0}$, $\displaystyle rank[\vec{v_{1}} ... \vec{v_{m}}]$ will fail to equal m. There will not be a pivot in every column, which is a requirement for a set of linearly independent vectors.
I mean, if you want to do it in terms of matrices you're correct, but isn't it just simpler to note that $\displaystyle 0v_1+\cdots+v_m=0$?