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27 lines
674 B
Markdown
27 lines
674 B
Markdown
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cards-deck: default_obsidian
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---
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#math/linear_algebra
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# definition
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A system of linear equations is said to be ==homogeneous== if it can be written as an augmented matrix of the form $[A \mid \mathbf{0}]$, where $A$ is an $m \times n$ matrix and $\mathbf{0}$ is the zero vector in $\mathbb{R}^{m}$.
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^1652734750442
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Example: The following system of equations is homogeneous (all of the constants on the right-hand side are zero):
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$$
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\begin{array}{r}
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3 x_{1}+5 x_{2}-4 x_{3}=0 \\
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-3 x_{1}-2 x_{2}+4 x_{3}=0 \\
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6 x_{1}+x_{2}-8 x_{3}=0
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\end{array}
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$$
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An example matrix:
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$\left[\begin{array}{ccc|c}
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3 & 5 & -4 & 0 \\
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-3 & -2 & 4 & 0 \\
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6 & 1 & -8 & 0
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\end{array}\right]$ |