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Exercise I.C.2

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We are interested in the fact whether there is a non-trivial linear combination

a [1, 1, 0] + b [2, 0, - 1] + c [- 3, 1, 1] = 0

In other words, whether the system of linear equations

{\begin{matrix} a + 2 b - 3 c & = 0 \\ a + c & = 0 \\ - b + c & = 0 \end{matrix}

has non-zero solution. We obviously have $a = - c$ and $b = c$ which contradicts to $- c + 2 c - 3 c = 0$ , i.e., $a = b = c = 0$ . Thus, $S$ is linearly independent. Since $\dim (V) = \dim (S)$ we see that $S$ is a basis for $ℝ^{3}$ .

togrul.topal

2021-10-30 12:13

Exercise I.C.2

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