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Linearly Independency of General Vectors

Linearly Independency of General Vectors

Definition

Let $V$ be a vector space over a scalar field $K$. Let $=\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$ be a set of vectors in $V$.

  1. We say that $S$ is linearly independent if whenever $c_1\mathbf{v}_1+\cdots+c_k\mathbf{v}_k=\mathbf{0}$ we have $c_1=\cdots=c_k=0$.

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Problems

  1. Let $C[3, 10]$ be the vector space consisting of all continuous functions defined on the interval $[3, 10]$. Consider the set $S=\{ \sqrt{x}, x^2 \}$ in $C[3,10]$. Show that the set $S$ is linearly independent in $C[3,10]$.

  2. Let $C[-\pi, \pi]$ be the vector space of all continuous functions defined on the interval $[-\pi, \pi]$. Show that the subset $\{\cos(x), \sin(x)\}$ in $C[-\pi, \pi]$ is linearly independent.

  3. Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$. Consider the functions $f(x)=\sin^2(x)$ and $g(x)=\cos^2(x)$ in $C[-2\pi, 2\pi]$. Prove or disprove that the functions $f(x)$ and $g(x)$ are linearly independent.
    (The Ohio State University)

  4. Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$. Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.
    (The Ohio State University)

  5. 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 $\R$.

  6. Let $V$ be an $n$-dimensional vector space over a field $K$.
    Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$. Are the following vectors linearly independent?
    \[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.\] If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.

  7. Let $V$ be a vector space over a scalar field $K$. Let $S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ be the set of vectors in $V$, where $n \geq 2$. Then prove that the set $S$ is linearly dependent if and only if at least one of the vectors in $S$ can be written as a linear combination of remaining vectors in $S$.

  8. 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 $\mathbf{u}_{n+1}\in V$. Show that $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n, \mathbf{u}_{n+1}$ are linearly independent if and only if $\mathbf{u}_{n+1} \not \in U$.

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