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Showing that two elements of a linearly independent Set Spans the same set

shen07

Member
Aug 14, 2013
54
Hi, i would like to have a hint for the following problem:

Let $$v_1, v_2 \&\ v_3 $$ in a vector space V over a field F such that$$ v_1+v_2+v_3=0$$, Show that $\{v_1,v_2\}$ spans the same subspace as $\{v_2,v_3\}$

Thanks in advance
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Hi, i would like to have a hint for the following problem:

Let $$v_1, v_2 \&\ v_3 $$ in a vector space V over a field F such that$$ v_1+v_2+v_3=0$$, Show that $\{v_1,v_2\}$ spans the same subspace as $\{v_2,v_3\}$

Thanks in advance
Note that $v_2$ and $v_3$ are both contained in $\text{span}(\{v_1,v_2\})$ (why?). Thus $\text{span}(\{v_2,v_3\})\subseteq \text{span}(\{v_1,v_2\})$.

Similarly $\text{span}(\{v_1,v_2\})\subseteq \text{span}(\{v_2,v_3\})$.

Therefore $\text{span}(\{v_1,v_2\})=\text{span}(\{v_2,v_3\})$.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,502
In general, if you have two sets of vectors $A$ and $B$ and every vector of $B$ is expressible through vectors of $A$, then $\mathop{\text{span}}(B)\subseteq \mathop{\text{span}}(A)$.