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Exercise 3.3.2

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We have

[\begin{matrix} 2 & 1 & 3 & b_{1} \\ 6 & 3 & 9 & b_{2} \\ 4 & 2 & 6 & b_{3} \end{matrix}] \to [\begin{matrix} 2 & 1 & 3 & b_{1} \\ 0 & 0 & 0 & b_{2} - 3 b_{1} \\ 0 & 0 & 0 & b_{3} - b_{1} \end{matrix}] .

Then,

[R d] = [\begin{matrix} 1 & 1 ∕ 2 & 3 ∕ 2 & 5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] .

$Ax = b$ has a solution when $b_{2} - 3 b_{1} = 0$ and $b_{3} - 2 b_{1} = 0;$ $C (A)$ line through $(2, 6, 4)$ which is the intersection of the planes $b_{2} - 3 b_{1} = 0$ and $b_{3} - 2 b_{1} = 0$ ; the nullspace contains all combinations of $s_{1} = (- 1 ∕ 2, 1, 0)$ and $s_{2} = (- 3 ∕ 2, 0, 1)$ ; particular solution $x_{p} = d = (5, 0, 0)$ and complete solution $x_{p} + c_{1} s_{1} + c_{2} s_{2}$ .

mathchakra

2022-01-23 14:28

Exercise 3.3.2

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