I am working through Serge Lang's undergraduate text: "Linear Algebra" and I've gotten hung up on a particular claim in one of his proofs on p. 15.
The result is:
Theorem 3.1: Let $V$ be a vector space over the field $K$. Let $\left\{ v_{1},\ldots,v_{m}\right\} $ be a basis of $V$ over $K$. Let $w_{1},\ldots,w_{n}$ be elements of $V$, and assume that $n>m$. Then $w_{1},\ldots,w_{n}$ are linearly dependent.
His proof starts out in the following way.
Proof: Assume that $w_{1},\ldots,w_{n}$ are linearly independent. Since $\left\{ v_{1},\ldots,v_{m}\right\} $ is a basis, there exists elements $a_{1},\ldots,a_{m}\in K$ such that $w_{1}=a_{1}v_{1}+\cdots+a_{m}v_{m} $.
And here is the part that I am having trouble with:
By assumption, we know that $w_{1}\neq O$ and hence some $a_{i}\ne0$.
I do not understand how he deduces that $w_{1}\neq O$. He has assumed that $w_{1},\ldots,w_{n}$ are linearly independent, which means that if we have a a linear combination of the form $a_{1}w_{1}+\cdots+a_{n}w_{n}=O$ then all of the coefficients $a_{1},\ldots,a_{m}$ must be zero. But I do not understand how to deduce from this that a particular $w_{i}\neq O$.
It seems like there should be a simple explanation, but I have not been able to puzzle it out.
Thanks in advance for the help!
