# Thread: Spanning and linear independence

1. ## Spanning and linear independence

Let v1,v2,...vn be linearly independent vectors in a vector space V. Show that v2,...vn cannot span V.

I'm confused because doesn't the linear independency proof contain the fact that say vectors v1,...vn are contained in Span(v1,...vn)? And then it is written uniquely as a linear combination of v1,...vn if and only if v1,...vn are linearly independent.

2. Originally Posted by pakman
Let v1,v2,...vn be linearly independent vectors in a vector space V. Show that v2,...vn cannot span V.

I'm confused because doesn't the linear independency proof contain the fact that say vectors v1,...vn are contained in Span(v1,...vn)? And then it is written uniquely as a linear combination of v1,...vn if and only if v1,...vn are linearly independent.
If $\displaystyle v_2,...,v_n$ spam $\displaystyle V$ then this set form a basis for $\displaystyle V$. We claim that $\displaystyle v_1,v_2,...,v_n$ cannot be linearly independent. Because since $\displaystyle v_2,...,v_n$ spam it means for any vector $\displaystyle v$ we can express it as a linear combination of those. Thus, $\displaystyle v_1 = k_2v_2+...+k_nv_n$. Thus, $\displaystyle v_1 - k_2v_2 -... - k_nv_n = 0$ not all the coefficients are zero, so it cannot be linearly independent.