Question

Show that the set of functions\(\left\{ {{\bf{1,}}\,{\bf{x,}}\,{{\bf{x}}^{\bf{2}}}{\bf{,}}...{\bf{,}}\,{{\bf{x}}^{\bf{n}}}} \right\}\), where n is a positive integer, is linearly independent on every open interval (a, b).

[Hint: Use the fact that a polynomial of degree at most n has no more than n zeros unless it is identically zero.]

Short answer

Thus, this set forms the linearly independent set on every open interval (a, b).

Step-by-step solution

Use the fact that a polynomial of degree at most n has no more than n zeros unless it is identically zero. 

A polynomial of degree n,

\({{\bf{a}}_{\bf{0}}}{\bf{1 + }}{{\bf{a}}_{\bf{1}}}{\bf{x + }}{{\bf{a}}_{\bf{2}}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}...{\bf{ + }}{{\bf{a}}_{\bf{n}}}{{\bf{x}}^{\bf{n}}}{\bf{ + }}...{\bf{ = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.....\left( {\bf{1}} \right)\)

Substitute \({\bf{x = 0}}\)in the above expression,

\({{\bf{a}}_{\bf{0}}}{\bf{ = 0}}\)

From equation (1),

\({{\bf{a}}_{\bf{1}}}{\bf{x + }}{{\bf{a}}_{\bf{2}}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}...{\bf{ + }}{{\bf{a}}_{\bf{n}}}{{\bf{x}}^{\bf{n}}}{\bf{ + }}...{\bf{ = 0}}\)

Take the derivative with respect to x,

\({{\bf{a}}_{\bf{1}}}{\bf{ + 2}}{{\bf{a}}_{\bf{2}}}{\bf{x + }}...{\bf{ + n}}{{\bf{a}}_{\bf{n}}}{{\bf{x}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....\left( {\bf{2}} \right)\)

Substitute \({\bf{x = 0}}\)in the above expression,

\({{\bf{a}}_{\bf{1}}}{\bf{ = 0}}\)

From equation (2),

\({\bf{2}}{{\bf{a}}_{\bf{2}}}{\bf{x + }}...{\bf{ + n}}{{\bf{a}}_{\bf{n}}}{{\bf{x}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ = 0}}\)

Take the derivative with respect to x,

\({\bf{2}}{{\bf{a}}_{\bf{2}}}{\bf{ + }}...{\bf{ + n}}\left( {{\bf{n - 1}}} \right){{\bf{a}}_{\bf{n}}}{{\bf{x}}^{{\bf{n - 2}}}}{\bf{ + }}...{\bf{ = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,....\left( {\bf{3}} \right)\)

Substitute \({\bf{x = 0}}\)in the above expression,

\({{\bf{a}}_{\bf{2}}}{\bf{ = 0}}\)

Conclusion.

Therefore, from this process, we can show that:

\({{\bf{a}}_{\bf{i}}}{\bf{ = 0}}\)

Thus, this set forms the linearly independent set on every open interval (a, b).