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Let $V$ be the space spanned by the two functions $\cos(t)$ and $\sin(t)$. Find the matrix $A$ of the linear transformation $T(f(t)) = f''(t) + 7f'(t) + 4f(t)$ from $V$ into itself with respect to the basis $\{\cos(t), \sin(t)\}$.

$$A = \left[\begin{matrix}\color{red}{\square} & \color{red}{\square} \\ \color{red}{\square} & \color{red}{\square} \end{matrix}\right]$$

I don't know where to start. Where does A go in regards to the $T(f(t))=\ldots$ equation?

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If this and the previous question are home work, assigned to you through the web, please add that tag. Don't worry, they will be answered. – user21436 Mar 28 '12 at 16:56
Search engines and visually impaired people (for instance) cannot read the text content of image files. It would be nice if you post your question in text. – André Caldas Mar 28 '12 at 17:04
Would be interesting to see, how it acts on the space of all $(\cos(nx),\sin(nx))$! – draks ... Mar 28 '12 at 18:26
up vote 3 down vote accepted

You're given a basis which I'll label as $\vec{v}_1 = \cos(t)$ and $\vec{v}_2 = \sin(t)$. This lets you interpret columns of numbers as vectors. The column vector $\begin{bmatrix} a\\b\end{bmatrix}$ literally means the vector $a\cos(t) + b\sin(t)$.

Writing the matrix $A$ relative to this basis is done by computing $A\vec{v}_1$ and expressing it in terms of $\vec{v}_1$ and $\vec{v}_2$ and then doing the same for $A\vec{v}_2$.

In matrix language, $T\vec{v}_1$ corresponds to $A\begin{bmatrix}1\\0\end{bmatrix}$. On the other hand, $T\vec{v}_1 = T\cos(t) = \cos(t)''+7\cos(t)' +4\cos(t) = 3\cos(t) -7\sin(t) = \begin{bmatrix}3\\-7\end{bmatrix}$

So, the matrix $A$ should have the property that $A\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix} 3\\-7\end{bmatrix}$. This forces $A$ to have the form $A = \begin{bmatrix} 3 & *\\-7 & *\end{bmatrix}$.

Can you take it from here?

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$\bf Hint:$ First determine where $T$ sends the basis $\{\cos(t),\sin(t)\}$. Then write the images as a linear combination of the basis and consider the matrix that has as columns the coefficients of the linear combination.

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