Determine whether the set B= {[1 0 1], [0 1 1], [1 1 0]} in R^3 is linearly independent.

Question

Determine whether the set [latex]B=\left\{\left [ \begin{matrix} 1 \ 0 \ 1 \end{matrix} ight ],\left [ \begin{matrix} 0 \ 1 \ 1 \end{matrix} ight ] ,\left [ \begin{matrix} 1 \ 1 \ 0 \end{matrix} ight ] ight\} [/latex] in [latex]\mathbb{R} ^{3}[/latex] is linearly independent.

Accepted Answer

By definition, we need to find all solutions of the equation

[latex]\left [ \begin{matrix} 0 \ 0 \ 0 \end{matrix} ight ] =c_{1}\left [ \begin{matrix} 1 \ 0 \ 1 \end{matrix} ight ] +c_{2}\left [ \begin{matrix} 0 \ 1 \ 1 \end{matrix} ight ] +c_{3}\left [ \begin{matrix} 1 \ 1 \ 0 \end{matrix} ight ] [/latex]

Performing the linear combination on the right-hand side gives

[latex]\left [ \begin{matrix} 0 \ 0 \ 0 \end{matrix} ight ] =\left [ \begin{matrix} c_{1}+c_{3} \ c_{2}+c_{3} \ c_{1}+c_{2} \end{matrix} ight ] [/latex]

Comparing entries gives the system of equations

[latex]c_{1}+c_{3}=0, \ \ \ \ \ \ \ \ c_{2}+c_{3}=0, \ \ \ \ \ \ \ \ c_{1}+c_{2}=0[/latex]

Adding the first to the second and then subtracting the third gives [latex]2c_{3}=0[/latex]. Hence, [latex]c_{3}=0[/latex] which then implies [latex]c_{1}=c_{2}=0[/latex] from the first and second equations. Since [latex]c_{1}=c_{2}=c_{3}=0[/latex]is the only solution, the set is linearly independent.

Question 1.2.6: Determine whether the set B= {[1 0 1], [0 1 1], [1 1 0]} in ...

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