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

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 $\mathbb{C}$

Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)

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