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

shen07

Member
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 $\{v_2,v_3\}$

caffeinemachine

Well-known member
MHB Math Scholar
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 $\{v_2,v_3\}$

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 $\text{span}(\{v_1,v_2\})=\text{span}(\{v_2,v_3\})$.
In general, if you have two sets of vectors $A$ and $B$ and every vector of $B$ is expressible through vectors of $A$, then $\mathop{\text{span}}(B)\subseteq \mathop{\text{span}}(A)$.