# Spanning and linear independence

• Nov 8th 2007, 06:04 PM
pakman
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.
• Nov 8th 2007, 06:20 PM
ThePerfectHacker
Quote:

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 $v_2,...,v_n$ spam $V$ then this set form a basis for $V$. We claim that $v_1,v_2,...,v_n$ cannot be linearly independent. Because since $v_2,...,v_n$ spam it means for any vector $v$ we can express it as a linear combination of those. Thus, $v_1 = k_2v_2+...+k_nv_n$. Thus, $v_1 - k_2v_2 -... - k_nv_n = 0$ not all the coefficients are zero, so it cannot be linearly independent.