Show that v1,v2,v3 are independent but v1,v2,v3,v4 are dependent: Solve c1v1+c2v2+c3v3+c4v4=0 or Ax = 0. The v’s go in

Question

Show that {v}_{1}, {v}_{2}, {v}_{3} are independent but {v}_{1}, {v}_{2}, {v}_{3}, {v}_{4} are dependent: {v}_{1}=\left[ \begin{matrix} 1 \ 0 \ 0 \end{matrix} ight] \quad {v}_{2}=\left[ \begin{matrix} 1 \ 1 \ 0 \end{matrix} ight] \quad {v}_{3}=\left[ \begin{matrix} 1 \ 1 \ 1 \end{matrix} ight] \quad {v}_{4}=\left[ \begin{matrix} 2 \ 3 \ 4 \end{matrix} ight]. Solve {c}_{1}{v}_{1} + {c}_{2}{v}_{2} + {c}_{3}{v}_{3} + {c}_{4}{v}_{4} = 0 or Ax = 0. The v's go in the columns of A.

Accepted Answer

[latex]\left[ \begin{matrix} 1 & 1 & 1 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{matrix} ight] \left[ \begin{matrix} { c }_{ 1 } \ { c }_{ 2 } \ { c }_{ 3 } \end{matrix} ight]=0[/latex] gives [latex]{ c }_{ 3 } = { c }_{ 2 } = { c }_{ 1 } = 0[/latex].

So those 3 column vectors are independent.

But [latex]\left[ \begin{matrix} 1 & 1 & 1 & 2 \ 0 & 1 & 1 & 3 \ 0 & 0 & 1 & 4 \end{matrix} ight] [c]=\left[ \begin{matrix} 0 \ 0 \ 0 \end{matrix} ight][/latex] is solved by c = (1, 1, -4, 1).

Then [latex]{v}_{1} + {v}_{2} - 4{v}_{3} + {v}_{4} = 0[/latex] (dependent).

Question : Show that v1,v2,v3 are independent but v1,v2,v3,v4 are depe...