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Mathematics LibreTexts

6.5: Review Problems

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    1. Show that the pair of conditions:

    \[(1)~~~\left\{\begin{matrix}L(u+v) = L(u) + L(v)\\L(cv) = cL(v)\end{matrix}\right.\]

    (valid for all vectors \(u,v\) and any scalar \(c\)) is equivalent to the single condition:
    $$(2)~~~L(ru + sv) = rL(u) + sL(v)$$
    (for all vectors \(u,v\) and any scalars \(r\) and \(s\)).
    Your answer should have two parts. Show that (1) \(\Rightarrow\) (2), and then show that (2) \(\Rightarrow\) (1),

    2. If \(f\) is a linear function of one variable, then how many points on the graph of the function are needed to specify the function? Give an explicit expression for \(f\) in terms of these points.

    a) If \(p\begin{pmatrix}1\\2\end{pmatrix}=1\) and \(p\begin{pmatrix}2\\4\end{pmatrix}=3\) is it possible that \(p\) is a linear function?

    b) If \(Q(x^{2})=x^{3}\) and \(Q(2x^{2})=x^{4}\) is it possible that \(Q\) is a linear function from polynomials to polynomials?

    4. If \(f\) is a linear function such that
    $$f\begin{pmatrix}1\\2\end{pmatrix}=0{\rm ,~and~} f\begin{pmatrix}2\\3\end{pmatrix}=1\, ,$$
    then what is \(f\begin{pmatrix}x\\y\end{pmatrix}\)?

    5. Let \(P_{n}\) be the space of polynomials of degree \(n\) or less in the variable \(t\). Suppose \(L\) is a linear transformation from \(P_{2} \rightarrow P_{3}\) such that \(L(1) = 4\), \(L(t)=t^{3}\), and \(L(t^{2}) = t-1\).

    a) Find \(L(1+t+2t^{2})\).

    b) Find \(L(a+bt+ct^{2})\).

    c) Find all values \(a,b,c\) such that \(L(a+bt+ct^{2})=1+3t+2t^{3}\).

    6. Show that the operator \(\cal{I}\) that maps \(f\) to the function \(\cal{I}f\) defined by \(\cal{I}f(x):=\int_{0}^{x}f(t)dt\) is a linear operator on the space of continuous functions.

    7. Let \(z \in \mathbb{C}\). Recall that we can express \(z = x + iy\) where \(x,y \in \mathbb{R}\), and we can form the \(\textit{complex conjugate} of \(z\) by taking \(\overline{z} = x - iy\). The function \(c \colon \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) which sends \((x, y) \mapsto (x, -y)\) agrees with complex conjugation.

    a) Show that \(c\) is a linear map over \(\mathbb{R}\) (\(\textit{i.e.}\) scalars in \(\mathbb{R}\)).

    b) Show that \(\overline{z}\) is not linear over \(\mathbb{C}\)


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