Question

In the vector space for all real-valued functions, find a basis for the subspace spanned by $\left\{\mathbf{s}\mathbf{i}\mathbf{n}t,\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{2}t,\mathbf{s}\mathbf{i}\mathbf{n}t\mathbf{c}\mathbf{o}\mathbf{s}t\right\}$.

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:

$\begin{array}{c}V=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{v_1,v_2,v_3\right\}\\ =\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{v_1,v_2\right\}\mathrm{o}\mathrm{r}\\ =\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\left\{v_1,v_3\right\}\end{array}$

So, the above set represents the basis of H.

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