Describe Span \left\{\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ] ,\left [ \begin{matrix} 6 \\ 4 \end{matrix} \right ] \right\} geometrically.
Question 1.2.4: Describe Span {[3 2], [6 4]} geometrically.
Using the definition of span, a vector equation of the spanned set is
\vec{x}=s\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ] +t\left [ \begin{matrix} 6 \\ 4 \end{matrix} \right ],\ \ \ \ s,t\in \mathbb{R}Observe that we can rewrite this as
\vec{x}=s\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ]+ (2t)\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ],\ \ \ \ s,t\in \mathbb{R}= (s+2t)\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ] , \ \ \ \ s,t \in \mathbb{R}
Since c = s + 2t can take any real value, the spanned set is a line through the origin with direction vector \left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ]
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