Exercise 3.4.1

Answers

The system

[\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}] = 0

gives $c_{3} = c_{2} = c_{1} = 0 .$ So those 3 column vectors are independent. But

[\begin{matrix} 1 & 1 & 1 & 2 \\ 0 & 1 & 1 & 3 \\ 0 & 0 & 1 & 4 \end{matrix}] [\begin{matrix} c \\ 0 \\ 0 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]

is solved by $c = (1, 1, - 4, 1) .$ Then $v_{1} + v_{2} - 4 v_{3} + v_{4} =$ dependent.

mathchakra

2022-01-23 15:41