I know that set of linearly independent vectors in $V$ will also be linearly independent in $U\subset V$. Can it be proven the same goes in reverse: Any linearly independent elements of $U$, when considered in the super-set V are always linearly independent?
Are all of the elements of a linearly independent set always linearly independent in the super set?
0
$\begingroup$
linear-algebra
proof-writing
1 Answers
0
Yes.
Linear independence in $U$ for say the set $u,v,w$ means $au+bv+cw =0_U$ exactly when $a=b=c=0$ (that's the zero scalar).
Linear independence in $V$ for the same set $u,v,w$ means $au+bv+cw =0_V$ exactly when $a=b=c=0$.
The two are equivalent because $U$ and $V$ always have the same identity vector if one contains the other. In other words $0_U=0_V$.