vault backup: 2026-10-05 09:13:05
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"snippets": "export default [\n // Math mode\n\t{trigger: \"mk\", replacement: \"$$0$\", options: \"tA\"},\n {trigger: \"dm\", replacement: \"$$\\n\\t$0\\n$$\", options: \"tAw\"},\n\t{trigger: /(?<=\\S.*)dm/, replacement: \"\\n$$\\n\\t$0\\n$$\", options: \"tAw\", priority: 1},\n\n\t{trigger: /([^\\\\])beg/, replacement: \"[[0]]\\\\begin{$0}\\n\\t$1\\n\\\\end{$0}\", options: \"MA\"},\n\t{trigger: /([^\\\\])beg/, replacement: \"[[0]]\\\\begin{$0} $1 \\\\end{$0}\", options: \"nA\"},\n\n // Dashes\n\t// {trigger: \"--\", replacement: \"–\", options: \"tA\"},\n\t// {trigger: \"–-\", replacement: \"—\", options: \"tA\"},\n\t// {trigger: \"—-\", replacement: \"---\", options: \"tA\"},\n\n // Greek letters\n\t{trigger: \"@a\", replacement: \"\\\\alpha\", options: \"mA\"},\n\t{trigger: \"@b\", replacement: \"\\\\beta\", options: \"mA\"},\n\t{trigger: \"@g\", replacement: \"\\\\gamma\", options: \"mA\"},\n\t{trigger: \"@G\", replacement: \"\\\\Gamma\", options: \"mA\"},\n\t{trigger: \"@d\", replacement: \"\\\\delta\", options: \"mA\"},\n\t{trigger: \"@D\", replacement: \"\\\\Delta\", options: \"mA\"},\n\t{trigger: \"@e\", replacement: \"\\\\epsilon\", options: \"mA\"},\n\t{trigger: \":e\", replacement: \"\\\\varepsilon\", options: \"mA\"},\n\t{trigger: \"@z\", replacement: \"\\\\zeta\", options: \"mA\"},\n\t{trigger: \"@t\", replacement: \"\\\\theta\", options: \"mA\"},\n\t{trigger: \"@T\", replacement: \"\\\\Theta\", options: \"mA\"},\n\t{trigger: \":t\", replacement: \"\\\\vartheta\", options: \"mA\"},\n\t{trigger: \"@i\", replacement: \"\\\\iota\", options: \"mA\"},\n\t{trigger: \"@k\", replacement: \"\\\\kappa\", options: \"mA\"},\n\t{trigger: \"@l\", replacement: \"\\\\lambda\", options: \"mA\"},\n\t{trigger: \"@L\", replacement: \"\\\\Lambda\", options: \"mA\"},\n\t{trigger: \"@s\", replacement: \"\\\\sigma\", options: \"mA\"},\n\t{trigger: \"@S\", replacement: \"\\\\Sigma\", options: \"mA\"},\n\t{trigger: \"@u\", replacement: \"\\\\upsilon\", options: \"mA\"},\n\t{trigger: \"@U\", replacement: \"\\\\Upsilon\", options: \"mA\"},\n\t{trigger: \"@o\", replacement: \"\\\\omega\", options: \"mA\"},\n\t{trigger: \"@O\", replacement: \"\\\\Omega\", options: \"mA\"},\n\t{trigger: \"ome\", replacement: \"\\\\omega\", options: \"mA\"},\n\t{trigger: \"Ome\", replacement: \"\\\\Omega\", options: \"mA\"},\n\n // Text environment\n {trigger: \"text\", replacement: \"\\\\text{$0}$1\", options: \"mA\"},\n {trigger: \"\\\"\", replacement: \"\\\\text{$0}$1\", options: \"mA\"},\n\n // Basic operations\n {trigger: \"sr\", replacement: \"^{2}\", options: \"mA\"},\n\t{trigger: \"cb\", replacement: \"^{3}\", options: \"mA\"},\n\t{trigger: \"rd\", replacement: \"^{$0}$1\", options: \"mA\"},\n\t{trigger: \"_\", replacement: \"_{$0}$1\", options: \"mA\"},\n\t{trigger: \"sts\", replacement: \"_\\\\text{$0}\", options: \"mA\"},\n\t{trigger: \"sq\", replacement: \"\\\\sqrt{ $0 }$1\", options: \"mA\"},\n\t{trigger: \"//\", replacement: \"\\\\frac{$0}{$1}$2\", options: \"mA\"},\n\t{trigger: \"ee\", replacement: \"e^{ $0 }$1\", options: \"mA\"},\n {trigger: \"invs\", replacement: \"^{-1}\", options: \"mA\"},\n\n {trigger: /([^\\\\])(exp|log|ln)/, replacement: \"[[0]]\\\\[[1]]\", options: \"rmA\"},\n {trigger: \"conj\", replacement: \"^{*}\", options: \"mA\"},\n {trigger: \"Re\", replacement: \"\\\\mathrm{Re}\", options: \"mA\"},\n\t{trigger: \"Im\", replacement: \"\\\\mathrm{Im}\", options: \"mA\"},\n {trigger: \"bf\", replacement: \"\\\\mathbf{$0}\", options: \"mA\"},\n\t{trigger: \"rm\", replacement: \"\\\\mathrm{$0}$1\", options: \"mA\"},\n\n // Linear algebra\n {trigger: /([^\\\\])(det)/, replacement: \"[[0]]\\\\[[1]]\", options: \"rmA\"},\n {trigger: \"trace\", replacement: \"\\\\mathrm{Tr}\", options: \"mA\"},\n\n // More operations\n\t{trigger: \"([a-zA-Z])hat\", replacement: \"\\\\hat{[[0]]}\", options: \"rmA\"},\n {trigger: \"([a-zA-Z])bar\", replacement: \"\\\\bar{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])dot\", replacement: \"\\\\dot{[[0]]}\", options: \"rmA\", priority: -1},\n\t{trigger: \"([a-zA-Z])ddot\", replacement: \"\\\\ddot{[[0]]}\", options: \"rmA\", priority: 1},\n\t{trigger: \"([a-zA-Z])tilde\", replacement: \"\\\\tilde{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])und\", replacement: \"\\\\underline{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])vec\", replacement: \"\\\\vec{[[0]]}\", options: \"rmA\"},\n {trigger: \"([a-zA-Z]),\\\\.\", replacement: \"\\\\mathbf{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])\\\\.,\", replacement: \"\\\\mathbf{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"\\\\\\\\(${GREEK}),\\\\.\", replacement: \"\\\\boldsymbol{\\\\[[0]]}\", options: \"rmA\"},\n\t{trigger: \"\\\\\\\\(${GREEK})\\\\.,\", replacement: \"\\\\boldsymbol{\\\\[[0]]}\", options: \"rmA\"},\n\n\t{trigger: \"hat\", replacement: \"\\\\hat{$0}$1\", options: \"mA\"},\n {trigger: \"bar\", replacement: \"\\\\bar{$0}$1\", options:Line truncated
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"snippets": "export default [\n // Math mode\n\t{trigger: \"mk\", replacement: \"$$0$\", options: \"tA\"},\n {trigger: \"dm\", replacement: \"$$\\n\\t$0\\n$$\", options: \"tAw\"},\n\t{trigger: /(?<=\\S.*)dm/, replacement: \"\\n$$\\n\\t$0\\n$$\", options: \"tAw\", priority: 1},\n\n\t{trigger: /([^\\\\])beg/, replacement: \"[[0]]\\\\begin{$0}\\n\\t$1\\n\\\\end{$0}\", options: \"MA\"},\n\t{trigger: /([^\\\\])beg/, replacement: \"[[0]]\\\\begin{$0} $1 \\\\end{$0}\", options: \"nA\"},\n\n // Dashes\n\t// {trigger: \"--\", replacement: \"–\", options: \"tA\"},\n\t// {trigger: \"–-\", replacement: \"—\", options: \"tA\"},\n\t// {trigger: \"—-\", replacement: \"---\", options: \"tA\"},\n\n // Greek letters\n\t{trigger: \"@a\", replacement: \"\\\\alpha\", options: \"mA\"},\n\t{trigger: \"@b\", replacement: \"\\\\beta\", options: \"mA\"},\n\t{trigger: \"@g\", replacement: \"\\\\gamma\", options: \"mA\"},\n\t{trigger: \"@G\", replacement: \"\\\\Gamma\", options: \"mA\"},\n\t{trigger: \"@d\", replacement: \"\\\\delta\", options: \"mA\"},\n\t{trigger: \"@D\", replacement: \"\\\\Delta\", options: \"mA\"},\n\t{trigger: \"@e\", replacement: \"\\\\epsilon\", options: \"mA\"},\n\t{trigger: \":e\", replacement: \"\\\\varepsilon\", options: \"mA\"},\n\t{trigger: \"@z\", replacement: \"\\\\zeta\", options: \"mA\"},\n\t{trigger: \"@t\", replacement: \"\\\\theta\", options: \"mA\"},\n\t{trigger: \"@T\", replacement: \"\\\\Theta\", options: \"mA\"},\n\t{trigger: \":t\", replacement: \"\\\\vartheta\", options: \"mA\"},\n\t{trigger: \"@i\", replacement: \"\\\\iota\", options: \"mA\"},\n\t{trigger: \"@k\", replacement: \"\\\\kappa\", options: \"mA\"},\n\t{trigger: \"@l\", replacement: \"\\\\lambda\", options: \"mA\"},\n\t{trigger: \"@L\", replacement: \"\\\\Lambda\", options: \"mA\"},\n\t{trigger: \"@s\", replacement: \"\\\\sigma\", options: \"mA\"},\n\t{trigger: \"@S\", replacement: \"\\\\Sigma\", options: \"mA\"},\n\t{trigger: \"@u\", replacement: \"\\\\upsilon\", options: \"mA\"},\n\t{trigger: \"@U\", replacement: \"\\\\Upsilon\", options: \"mA\"},\n\t{trigger: \"@o\", replacement: \"\\\\omega\", options: \"mA\"},\n\t{trigger: \"@O\", replacement: \"\\\\Omega\", options: \"mA\"},\n\t{trigger: \"ome\", replacement: \"\\\\omega\", options: \"mA\"},\n\t{trigger: \"Ome\", replacement: \"\\\\Omega\", options: \"mA\"},\n\n // Text environment\n {trigger: \"text\", replacement: \"\\\\text{$0}$1\", options: \"mA\"},\n {trigger: \"\\\"\", replacement: \"\\\\text{$0}$1\", options: \"mA\"},\n\n // Basic operations\n {trigger: \"sr\", replacement: \"^{2}\", options: \"mA\"},\n\t{trigger: \"cb\", replacement: \"^{3}\", options: \"mA\"},\n\t{trigger: \"rd\", replacement: \"^{$0}$1\", options: \"mA\"},\n\t{trigger: \"_\", replacement: \"_{$0}$1\", options: \"mA\"},\n\t{trigger: \"sts\", replacement: \"_\\\\text{$0}\", options: \"mA\"},\n\t{trigger: \"sq\", replacement: \"\\\\sqrt{ $0 }$1\", options: \"mA\"},\n\t{trigger: \"//\", replacement: \"\\\\frac{$0}{$1}$2\", options: \"mA\"},\n\t{trigger: \"ee\", replacement: \"e^{ $0 }$1\", options: \"mA\"},\n {trigger: \"invs\", replacement: \"^{-1}\", options: \"mA\"},\n\n {trigger: /([^\\\\])(exp|log|ln)/, replacement: \"[[0]]\\\\[[1]]\", options: \"rmA\"},\n {trigger: \"conj\", replacement: \"^{*}\", options: \"mA\"},\n {trigger: \"Re\", replacement: \"\\\\mathrm{Re}\", options: \"mA\"},\n\t{trigger: \"Im\", replacement: \"\\\\mathrm{Im}\", options: \"mA\"},\n {trigger: \"bf\", replacement: \"\\\\mathbf{$0}\", options: \"mA\"},\n\t{trigger: \"rm\", replacement: \"\\\\mathrm{$0}$1\", options: \"mA\"},\n\n // Linear algebra\n {trigger: /([^\\\\])(det)/, replacement: \"[[0]]\\\\[[1]]\", options: \"rmA\"},\n {trigger: \"trace\", replacement: \"\\\\mathrm{Tr}\", options: \"mA\"},\n\n // More operations\n\t{trigger: \"([a-zA-Z])hat\", replacement: \"\\\\hat{[[0]]}\", options: \"rmA\"},\n {trigger: \"([a-zA-Z])bar\", replacement: \"\\\\bar{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])dot\", replacement: \"\\\\dot{[[0]]}\", options: \"rmA\", priority: -1},\n\t{trigger: \"([a-zA-Z])ddot\", replacement: \"\\\\ddot{[[0]]}\", options: \"rmA\", priority: 1},\n\t{trigger: \"([a-zA-Z])tilde\", replacement: \"\\\\tilde{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])und\", replacement: \"\\\\underline{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])vec\", replacement: \"\\\\vec{[[0]]}\", options: \"rmA\"},\n {trigger: \"([a-zA-Z]),\\\\.\", replacement: \"\\\\mathbf{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"([a-zA-Z])\\\\.,\", replacement: \"\\\\mathbf{[[0]]}\", options: \"rmA\"},\n\t{trigger: \"\\\\\\\\(${GREEK}),\\\\.\", replacement: \"\\\\boldsymbol{\\\\[[0]]}\", options: \"rmA\"},\n\t{trigger: \"\\\\\\\\(${GREEK})\\\\.,\", replacement: \"\\\\boldsymbol{\\\\[[0]]}\", options: \"rmA\"},\n\n\t{trigger: \"hat\", replacement: \"\\\\hat{$0}$1\", options: \"mA\"},\n {trigger: \"bar\", replacement: \"\\\\bar{$0}$1\", options:Line truncated
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"snippetVariables": "export default {\n\t\"${GREEK}\": \"(?:alpha|beta|gamma|Gamma|delta|Delta|epsilon|varepsilon|zeta|eta|theta|vartheta|Theta|iota|kappa|lambda|Lambda|mu|nu|xi|omicron|pi|rho|varrho|sigma|Sigma|tau|upsilon|Upsilon|phi|varphi|Phi|chi|psi|omega|Omega)\",\n\t\"${SYMBOL}\": \"(?:parallel|perp|partial|nabla|hbar|ell|infty|oplus|ominus|otimes|oslash|square|star|dagger|vee|wedge|subseteq|subset|supseteq|supset|emptyset|exists|nexists|forall|implies|impliedby|iff|setminus|neg|lor|land|bigcup|bigcap|cdot|times|simeq|approx)\",\n\t\"${MORE_SYMBOLS}\": \"(?:leq|geq|neq|gg|ll|equiv|sim|propto|rightarrow|leftarrow|Rightarrow|Leftarrow|leftrightarrow|to|mapsto|cap|cup|in|sum|prod|exp|ln|log|det|dots|vdots|ddots|pm|mp|int|iint|iiint|oint)\",\n \"${ACCENT}\": \"(?:dot|ddot|hat|bar|tilde|vec|underline|overline|mathbf|mathcal|mathrm|mathbb)\",\n}\n",
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"snippetVariables": "export default {\n\t\"${GREEK}\": \"(?:alpha|beta|gamma|Gamma|delta|Delta|epsilon|varepsilon|zeta|eta|theta|vartheta|Theta|iota|kappa|lambda|Lambda|mu|nu|xi|omicron|pi|rho|varrho|sigma|Sigma|tau|upsilon|Upsilon|phi|varphi|Phi|chi|psi|omega|Omega)\",\n\t\"${SYMBOL}\": \"(?:parallel|perp|partial|nabla|hbar|ell|infty|oplus|ominus|otimes|oslash|square|star|dagger|vee|wedge|subseteq|subset|supseteq|supset|emptyset|exists|nexists|forall|implies|impliedby|iff|setminus|neg|lor|land|bigcup|bigcap|cdot|times|simeq|approx)\",\n\t\"${MORE_SYMBOLS}\": \"(?:leq|geq|neq|gg|ll|equiv|sim|propto|rightarrow|leftarrow|Rightarrow|Leftarrow|leftrightarrow|to|mapsto|cap|cup|in|sum|prod|exp|ln|log|det|dots|vdots|ddots|pm|mp|int|iint|iiint|oint)\",\n \"${ACCENT}\": \"(?:dot|ddot|hat|bar|tilde|vec|underline|overline|mathbf|mathcal|mathrm|mathbb)\",\n}\n",
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"snippetsEnabled": true,
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"snippetsEnabled": true,
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"snippetsTrigger": "Tab",
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"snippetsTrigger": "Tab",
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@@ -8,9 +8,9 @@
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- $\sim$ is used to indicate the transformation between equivalent linear systems
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- $\sim$ is used to indicate the transformation between equivalent linear systems
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- matrices can be used to simplify when working with systems of linear equations
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- matrices can be used to simplify when working with systems of linear equations
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- matrices with all constant terms of a linear system of equations are called a **augmented matrix**
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- matrices with all constant terms of a linear system of equations are called a **augmented matrix**
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- $$\begin{gathered} \text{Linear System} \\ a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} = b_{2} \\ a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} = b_{3} \end{gathered} \quad \sim \quad \begin{gathered} \text{Augmented Matrix} \\ \begin{bmatrix}
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- $$\begin{gathered} \text{Linear System} \\ a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} = b_{2} \\ a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} = b_{3} \end{gathered} \quad \sim \quad \begin{gathered} \text{Augmented Matrix} \\ \left[\begin{array}{ccc|c}
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a_{11} & a_{12} & a_{13} & b_{1} \\ a_{21} & a_{22} & a_{23} & b_{2} \\ a_{31} & a_{32} & a_{33} & b_{3}
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a_{11} & a_{12} & a_{13} & b_{1} \\ a_{21} & a_{22} & a_{23} & b_{2} \\ a_{31} & a_{32} & a_{33} & b_{3}
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\end{bmatrix} \end{gathered}$$
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\end{array}\right] \end{gathered}$$
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- Same elementary operations can be used with augmented matrices, now it is with rows instead of equations.
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- Same elementary operations can be used with augmented matrices, now it is with rows instead of equations.
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- Gaussian Elimination
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- Gaussian Elimination
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- this is converting a matrix to **echelon form** (or row echelon form)
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- this is converting a matrix to **echelon form** (or row echelon form)
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@@ -32,21 +32,15 @@
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To find when the system $$\begin{gathered} 3x_{1} + 6x_{2} = 2 \\ hx_{1} + 12x_{2} = k \end{gathered}$$ has no solutions
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To find when the system $$\begin{gathered} 3x_{1} + 6x_{2} = 2 \\ hx_{1} + 12x_{2} = k \end{gathered}$$ has no solutions
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$$\begin{bmatrix}
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$$\left[\begin{array}{cc|c}
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3 & 6 & 2 \\ h & 12 & k
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3 & 6 & 2 \\ h & 12 & k
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\end{bmatrix} \sim \begin{bmatrix}
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\end{array}\right] \sim \left[\begin{array}{cc|c}
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h & 2h & \frac{2h}{3} \\ h & 12 & k
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h & 2h & \frac{2h}{3} \\ h & 12 & k
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\end{bmatrix} \sim \left[\begin{array}{cc|c}
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\end{array}\right] \sim \left[\begin{array}{cc|c}
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h & 2h & \frac{2h}{3} \\ 0 & 12 - 2h & k - \frac{2h}{3}
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h & 2h & \frac{2h}{3} \\ 0 & 12 - 2h & k - \frac{2h}{3}
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\end{array}\right] $$
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\end{array}\right] $$
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There are no solutions the system of equations when there is no pivot in a row and the value on the right is not equal to zero (if the value on the right is also equal to zero than that equation is essentially $0=0$ which is always true).
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There are no solutions the system of equations when there is no pivot in a row and the value on the right is not equal to zero (if the value on the right is also equal to zero than that equation is essentially $0=0$ which is always true).
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This means that there are no solutions when $$12-2h = 0 \to h=6$$ and $$k-\frac{2h}{3} \ne 0 \to k-4 \ne 0 \to k \ne 4$$
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This means that there are no solutions when $$12-2h = 0 \to h=6$$ and $$ \begin{gather} k-\frac{2h}{3} \ne 0 \\ k-4 \ne 0 \\[1.5ex] k \ne 4 \end{gather}$$
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$$
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\left[
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\begin{array}{cccc|c}
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1 & 0 & 3 & -1 & 0 \\
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0 & 1 & 1 & -1 & 0 \\
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0 & 0 & 0 & 0 & 0 \\
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\end{array}
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\right]
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$$
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