Describe Span {[3 1 -3], [0 -1 1], [3 0 -2]} geometrically.

Question

Describe Span [latex]\left\{\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] ,\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] ,\left [ \begin{matrix} 3 \ 0 \ -2 \end{matrix} ight ] ight\}[/latex] geometrically.

Accepted Answer

By definition, a vector equation for the spanned set is

[latex]\vec{x}=c_{1}\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +c_{2}\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] +c_{3}\left [ \begin{matrix} 3 \ 0 \ -2 \end{matrix} ight ], \ \ \ \ c_{1},c_{2},c_{3}\in \mathbb{R} [/latex]

We observe that [latex]\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] =\left [ \begin{matrix} 3 \ 0 \ -2 \end{matrix} ight ] [/latex] . Hence, we can rewrite the vector equation as

[latex]\vec{x}=c_{1}\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +c_{2}\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] +c_{3}\left\lgroup\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] ight group , \ \ \ c_{1},c_{2},c_{3}\in \mathbb{R} [/latex]

[latex]=(c_{1}+c_{3})\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +(c_{2}+c_{3})\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] , \ \ \ c_{1},c_{2},c_{3}\in \mathbb{R} [/latex]

Since [latex]\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ][/latex] and [latex]\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] [/latex] are not scalar multiples of each other, we cannot simplify the vector equation any more. Thus, the set is a plane with vector equation

[latex]\vec{x}=s\left [ \begin{matrix} 3 \ 1 \ -3 \end{matrix} ight ] +t\left [ \begin{matrix} 0 \ -1 \ 1 \end{matrix} ight ] , \ \ \ s,t\in \mathbb{R} [/latex]

Question 1.2.5: Describe Span {[3 1 -3], [0 -1 1], [3 0 -2]} geometrically.

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