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# 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.

3.
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 $$L(a+bt+ct^{2})=1+3t+2t^{3}$$.
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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 $$\mathbb{C}$$