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

The Wronskian for the set $\{e^x, e^{2x}, e^{3x}\}$ is given by
\[W(x):=\begin{vmatrix}
e^x & e^{2x} & e^{3x} \\
e^x &2e^{2x} &3e^{3x} \\
e^x & 4e^{2x} & 9e^{3x}
\end{vmatrix}.\]

Exponential Functions Form a Basis of a Vector Space
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) […]

Exponential Functions are Linearly Independent
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 $\R$.
Hint.
Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\]
[…]

The Matrix Exponential of a Diagonal Matrix
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 & […]

Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$
Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ […]

Every Basis of a Subspace Has the Same Number of Vectors
Let $V$ be a subspace of $\R^n$.
Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a basis of the subspace $V$.
Prove that every basis of $V$ consists of $k$ vectors in $V$.
Hint.
You may use the following fact:
Fact.
If […]

Linear Independent Vectors and the Vector Space Spanned By Them
Let $V$ be a vector space over a field $K$. Let $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ be linearly independent vectors in $V$. Let $U$ be the subspace of $V$ spanned by these vectors, that is, $U=\Span \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$.
Let […]

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