# Tagged: exponential function

## Problem 681

For a square matrix $M$, its matrix exponential is defined by
$e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.$

Suppose that $M$ is a diagonal matrix
$M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix}.$

Find the matrix exponential $e^M$.

## Problem 590

Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
$V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}$ be a subset in $C[-1, 1]$.

(a) Prove that $V$ is a subspace of $C[-1, 1]$.

(b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$.

(c) Prove that
$B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}$ is a basis for $V$.

## Problem 549

By calculating the Wronskian, determine whether the set of exponential functions
$\{e^x, e^{2x}, e^{3x}\}$ is linearly independent on the interval $[-1, 1]$.

## Problem 322

Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers.

(a) Prove that the map $\exp:\R \to \R^{\times}$ defined by
$\exp(x)=e^x$ is an injective group homomorphism.

(b) Prove that the additive group $\R$ is isomorphic to the multiplicative group
$\R^{+}=\{x \in \R \mid x > 0\}.$

## Problem 73

Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.

Show that exponential functions
$e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$.