Question

Suppose is a linearly dependent spanning set for a \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\)vector space V. Show that each w in \(V\) can be expressed in more than one way as a linear combination of \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}\).(Hint: Let \({\mathop{\rm w}\nolimits} = {k_1}{{\mathop{\rm v}\nolimits} _1} + ... + {k_4}{{\mathop{\rm v}\nolimits} _4}\) be an arbitrary vector in \(V\). Use the linear dependence of \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}} \right\}\) to produce another representation of w as a linear combination of \({{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _4}\).)

Short answer

It is proved that w in \(V\) can be expressed in more than one way as a linear combination of \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},\) and \({{\mathop{\rm v}\nolimits} _4}\).

Step-by-step solution

Show that each w in \(V\) can be expressed in more than one way as a linear combination

There are scalars \({k_1},{k_2},{k_3},\) and \({k_4}\) for \(w\)in \(V\) such that

\({\mathop{\rm w}\nolimits} = {k_1}{{\mathop{\rm v}\nolimits} _1} + {k_2}{{\mathop{\rm v}\nolimits} _2} + {k_3}{{\mathop{\rm v}\nolimits} _3} + {k_4}{{\mathop{\rm v}\nolimits} _4}\). …(1)

Since \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) spans \(V\) and the set is linearly dependent, scalars \({c_1},{c_2},{c_3},\) and \({c_4}\) are not all zero such that

\(0 = {c_1}{{\mathop{\rm v}\nolimits} _1} + {c_2}{{\mathop{\rm v}\nolimits} _2} + {c_3}{{\mathop{\rm v}\nolimits} _3} + {c_4}{{\mathop{\rm v}\nolimits} _4}\). …(2)

Add equations (1) and (2).

\(\begin{array}{c}{\mathop{\rm w}\nolimits} = {\mathop{\rm w}\nolimits} + 0\\ = \left( {{k_1} + {c_1}} \right){{\mathop{\rm v}\nolimits} _1} + \left( {{k_2} + {c_2}} \right){{\mathop{\rm v}\nolimits} _2} + \left( {{k_3} + {c_3}} \right){{\mathop{\rm v}\nolimits} _3} + \left( {{k_4} + {c_4}} \right){{\mathop{\rm v}\nolimits} _4}\end{array}\) …(3)

At least one of the weights in equation (3) is different from the corresponding weight in equation (1) since at least one of the \({c_i}\) is non-zero.

Therefore, \({\mathop{\rm w}\nolimits} \) can be expressed in more than one way as a linear combination of \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},\) and \({{\mathop{\rm v}\nolimits} _4}\).