6.7.1 Fantasy Basis (Elementary Standard Basis for R5

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Elementary standard basis for

R^{5}

Let

{\vec{e}}_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}], {\vec{e}}_{2} = [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{matrix}], {\vec{e}}_{3} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}], {\vec{e}}_{4} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{matrix}], {\vec{e}}_{5} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}]

Solution. Consider

\begin{aligned} x_{1} {\vec{e}}_{1} + x_{2} {\vec{e}}_{2} + \dots + x_{5} {\vec{e}}_{5} & = x_{1} [\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}] + x_{2} [\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}] + \dots + x_{5} [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{array}] \\ = [\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}] ⟺ all scalars are zero \end{aligned}

Therefore, the vectors are linearly independent.