Question

Explain what is wrong with the following discussion: Let \({\bf{f}}\left( t \right) = {\bf{3}} + t\) and \({\bf{g}}\left( t \right) = {\bf{3}}t + {t^{\bf{2}}}\), and note that \({\bf{g}}\left( t \right) = t{\bf{f}}\left( t \right)\). Then, \(\left\{ {{\bf{f}},{\bf{g}}} \right\}\) is linearly dependent because g is a multiple of f.

Short answer

t is not a scalar quantity.

Step-by-step solution

Write the given data

The functions \(f\left( t \right)\) and \(g\left( t \right)\) hold the property given below:

\(f\left( t \right) = tg\left( t \right)\)

\(\left\{ {{\bf{f}},{\bf{g}}} \right\}\)is linearly dependent because g is a multiple of f.

Check for linear dependency

If each vector in a set of two vectors is a scalar multiple of the other, or if any vector in the set is zero, the set is said to be linearly dependent.

In the equation \({\bf{g}}\left( t \right) = t{\bf{f}}\left( t \right)\), t is not a scalar quantity.

So, \({\bf{f}}\left( t \right)\)and \({\bf{g}}\left( t \right)\) are not linearly dependent.