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YCor
YCor
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Linear Independenceindependence of Elementelement-Wise Powerswise powers of Positive Vectorspositive vectors

Attempted to increase readability. + added higher order tag
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András Bátkai
András Bátkai
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Consider a vector $x$ with $0 < x_1 < \cdots < x_n < \infty$, and let $0 < \gamma_1 < \cdots < \gamma_n < \infty$. I would like to show that $x^{\gamma_1}, \ldots, x^{\gamma_n}$ are linearly independent, where $x^{\gamma_i}$ is defined as the vector $(x_1^{\gamma_i}, \ldots, x_n^{\gamma_i})$. It

I would like to show that $x^{\gamma_1}, \ldots, x^{\gamma_n}$ are linearly independent, where $x^{\gamma_i}$ is defined as the vector $(x_1^{\gamma_i}, \ldots, x_n^{\gamma_i})$.

It is clear that $x^{\gamma_1}$ and $x^{\gamma_2}$ are linearly independent, and I might have an overly complicated argument for the case $x^{\gamma_1}$, $x^{\gamma_2}$, and $x^{\gamma_3}$, but I'm really at a loss about how to tackle the general case.

Consider a vector $x$ with $0 < x_1 < \cdots < x_n < \infty$, and let $0 < \gamma_1 < \cdots < \gamma_n < \infty$. I would like to show that $x^{\gamma_1}, \ldots, x^{\gamma_n}$ are linearly independent, where $x^{\gamma_i}$ is defined as the vector $(x_1^{\gamma_i}, \ldots, x_n^{\gamma_i})$. It is clear that $x^{\gamma_1}$ and $x^{\gamma_2}$ are linearly independent, and I might have an overly complicated argument for the case $x^{\gamma_1}$, $x^{\gamma_2}$, and $x^{\gamma_3}$, but I'm really at a loss about how to tackle the general case.

Consider a vector $x$ with $0 < x_1 < \cdots < x_n < \infty$, and let $0 < \gamma_1 < \cdots < \gamma_n < \infty$.

I would like to show that $x^{\gamma_1}, \ldots, x^{\gamma_n}$ are linearly independent, where $x^{\gamma_i}$ is defined as the vector $(x_1^{\gamma_i}, \ldots, x_n^{\gamma_i})$.

It is clear that $x^{\gamma_1}$ and $x^{\gamma_2}$ are linearly independent, and I might have an overly complicated argument for the case $x^{\gamma_1}$, $x^{\gamma_2}$, and $x^{\gamma_3}$, but I'm really at a loss about how to tackle the general case.

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Trajmentor
Trajmentor
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Linear Independence of Element-Wise Powers of Positive Vectors

Consider a vector $x$ with $0 < x_1 < \cdots < x_n < \infty$, and let $0 < \gamma_1 < \cdots < \gamma_n < \infty$. I would like to show that $x^{\gamma_1}, \ldots, x^{\gamma_n}$ are linearly independent, where $x^{\gamma_i}$ is defined as the vector $(x_1^{\gamma_i}, \ldots, x_n^{\gamma_i})$. It is clear that $x^{\gamma_1}$ and $x^{\gamma_2}$ are linearly independent, and I might have an overly complicated argument for the case $x^{\gamma_1}$, $x^{\gamma_2}$, and $x^{\gamma_3}$, but I'm really at a loss about how to tackle the general case.