Section 2 Theorems

Theorem 1
If $\vec{u}_1.\vec{u}_2,...,\vec{u}_p$ are in a vectorspace V then the span{$\vec{v}_1,\vec{v}_2,...,\vec{v}_p$} is a subspace of V where span is defined as {$\vec{x}|c_1\vec{v}_1+c_2\vec{v}_2+...+c_p\vec{v}_p$} where $c \in \Re$

Proof
Let $\vec{v}_1,...,\vec{v}_p$ be a set of vectors in a Vectorspace V
Let H = span{$\vec{v}_1,...,\vec{v}_p$}
1) Let $c_i = 0$ for all i, then $0\vec{v}_1+ ... + 0\vec{v}_p = \vec{0}$
Hence, $\vec{0} \in H$
2) Let $\vec{u}_1$ and $\vec{u}_2$ \in H\vec{u}_1 = a_1\vec{v}_1 + ... + a_p\vec{v}_p = \sum\limits_{i=1}^p a_i\vec{v}_i\vec{u}_2 = b_1\vec{v}_1 + ... + b_p\vec{v}_p = \sum\limits_{i=1}^p b_i\vec{v}_i\vec{u}_1 + \vec{u}_2 = \sum\limits_{i=1}^p a_i\vec{v}_i + \sum\limits_{i=1}^p b_i\vec{v}_i = \sum\limits_{i=1}^p(a_i\vec{v}_i + b_i\vec{v}_i) = \sum\limits_{i=1}^p (c_i)\vec{v}_1$where$c_i = a_i+b_i$3)$c\vec{u} = c\sum\limits_{i=1}^p a_i\vec{v}_i = \sum\limits_{i=1}^p (ca_i)\vec{v}_i \in H\$

Hence, H is a subspace of V.

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