Question

In the vector space for all real-valued functions, find a basis for the subspace spanned by \(\left\{ {{\bf{sin}}t,\,{\bf{sin2}}t,\,{\bf{sin}}t\,{\bf{cos}}t} \right\}\).

Short answer

\(\left\{ {\sin t,\sin t\cos t} \right\}\)and \(\left\{ {\sin t,\;\sin 2t} \right\}\)

Step-by-step solution

Find the set of vectors for V

Let \({v_1} = \sin t\), \({v_2} = \sin 2t\), \({v_3} = \sin t\cos t\).

Simplify the equation \({v_3} = \sin t\cos t\) using trigonometric identities.

\[\begin{array}{c}{v_3} = \frac{1}{2}\left( {2\sin t\cos t} \right)\\ = \frac{1}{2}\sin 2t\\ = \frac{1}{2}{v_2}\end{array}\]

So, the vectors \({v_2}\) and \({v_3}\) are dependent.

Write the spanning set

The spanning set reduces as shown below:

\(\begin{array}{c}V = {\rm{span}}\left\{ {{v_1},\,{v_2},\,{v_3}} \right\}\\ = {\rm{span}}\left\{ {{v_1},{v_2}} \right\}\;\;{\rm{or}}\\ = {\rm{span}}\left\{ {{v_1},{v_3}} \right\}\end{array}\)

So, the above set represents the basis of H.

So, the basis are \(\left\{ {\sin t,\sin t\cos t} \right\}\) and \(\left\{ {\sin t,\;\sin 2t} \right\}\).