# Linear independence and span of two vectors

The definition of a linear independence is - A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the other vectors. If no vector in the set can be written in this way, then the vectors are said to be linearly independent.

I was given this question

a) Prove that span(v1,v2) belongs to P(i.e. any linear combination of v1 and v2 is on the plane)

I was also given that (v1,v2) is linearly independent.

So, how can i write these two vectors as a linear combination of each other if by the definition of linear independency no vectors can be written as a linear combination of other vectors.

• $v_1=1\cdot v_1+0\cdot v_2$ – DonQuixote Sep 27 '15 at 20:42
• @DonQuixote But then, v1 and v2 can be written as a linear combination of each other. Would it make (v1, v2) linear independent because they can be written as a linear combination. – Jack Sep 27 '15 at 21:02
• No, the set $\{v_1,v_2\}$ is said to be linearly independent iff $\forall \lambda_1,\lambda_2\in\mathbb{R}$, $\lambda_1v_1+\lambda_2v_2=\vec0$, then $\lambda_1=\lambda_2=0$. – DonQuixote Sep 27 '15 at 21:15
• $v_1\in span\{v_1,v_2\}$ always, it is a fact. It doesn't matter if $\{v_1,v_2\}$ is l.i. – DonQuixote Sep 27 '15 at 21:17
• @DonQuixote Oh i see, thank you so much ! Can you post your answer as a response but not comments? So i can accept your answer – Jack Sep 27 '15 at 21:23

No, the set $\{v1,v2\}$is said to be linearly independent iff $\forall\lambda_1,\lambda_2\in\mathbb R$ $\lambda_1v_1+\lambda_2v_2=\vec{0}$, then $\lambda_1=\lambda_2=0$.. The fact that $v_1\in span\{v_1,v_2\}$ is unrelated with $\{v_1,v_2\}$ is l.i.