Exercise 9.1

Question

Exercise 1: 
    If $S$ is a nonempty subset of a vector space $X$, prove that the span of $S$ is a vector space.

analambanomenos · Accepted Answer

Let $\v x,\v y$ be in the span of $S$ and let $c$ be a scalar; we need to show that $\v x+\v y$ and $c\v x$ are in the span of $S$. We have 
    \begin{align*}
        \v x &= c_1\v x_1+\cdots+c_m\v x_m \
        \v y &= d_1\v y_1+\cdots+d_n\v y_n
    \end{align*}
    for some $\v x_1,\ldots,\v x_m,\v y_1,\ldots,\v y_n$ in $S$ and some scalars $c_1,\ldots,c_m,d_1,\ldots,d_n$. Hence
    \begin{align*}
        \v x+\v y &= c_1\v x_1+\cdots+c_m\v x_m+d_1\v y_1+\cdots+d_n\v y_n \
        c\v x &= cc_1\v x_1+\cdots+cc_m\v x_m
    \end{align*}
    shows that $\v x+\v y$ and $c\v x$ are in the span of $S$.

Exercise 9.1

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Exercise 9.1

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