1. ## Set proof

Q: Prove that if $\displaystyle S_{1}$ is a nonempty subset of the finite set $\displaystyle S_{2}$, and $\displaystyle S_{1}$ is linearly dependent, then so is $\displaystyle S_{2}$

A: By definition $\displaystyle S_{1}\subset{S_{2}}$ $\displaystyle \iff$ $\displaystyle \forall{\vec{v}}\in{U}$$\displaystyle (\vec{v}\in{S_{1}} \displaystyle \Rightarrow \displaystyle \vec{v}\in{S_{2}}). Since we know every vector \displaystyle \vec{v_{i}}\in{S_{1}} can be written as a linear combination of at least one vector \displaystyle \vec{u}\in{S_{1}} and \displaystyle S_{1}\subset{S_{2}} we are guaranteed \displaystyle \vec{u}\in{S_2}. So, we have shown exists a linear dependent vector in \displaystyle S_{2}. \displaystyle \therefore \displaystyle S_{2} must also be linearly dependent. Have I shown enough work to prove my claim? 2. ## Linear dependence Hello Danneedshelp Originally Posted by Danneedshelp Q: Prove that if \displaystyle S_{1} is a nonempty subset of the finite set \displaystyle S_{2}, and \displaystyle S_{1} is linearly dependent, then so is \displaystyle S_{2} A: By definition \displaystyle S_{1}\subset{S_{2}} \displaystyle \iff \displaystyle \forall{\vec{v}}\in{U}$$\displaystyle (\vec{v}\in{S_{1}}$ $\displaystyle \Rightarrow$ $\displaystyle \vec{v}\in{S_{2}})$. Since we know every vector $\displaystyle \vec{v_{i}}\in{S_{1}}$ can be written as a linear combination of at least one vector $\displaystyle \vec{u}\in{S_{1}}$ and $\displaystyle S_{1}\subset{S_{2}}$ we are guaranteed $\displaystyle \vec{u}\in{S_2}$. So, we have shown exists a linear dependent vector in $\displaystyle S_{2}$.

$\displaystyle \therefore$ $\displaystyle S_{2}$ must also be linearly dependent.

Have I shown enough work to prove my claim?
This is OK, except that it's not every vector $\displaystyle \vec{v_{i}}\in{S_{1}}$, but $\displaystyle \exists\,\vec{v_{i}}\in{S_{1}}$.