Anki Deck Changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: Au,kdh[/@(

modified

Before

Front

A safe edge is a edge that is included in at all MSTs.

Back

A safe edge is a edge that is included in at all MSTs.

all, If the edge-weights are distinct, which means there is one unique MST.

After

Front

A safe edge is an edge that is included in at all MSTs.

Back

A safe edge is an edge that is included in at all MSTs.

all, If the edge-weights are distinct, which means there is one unique MST.

Field-by-field Comparison

Field	Before	After
Text	A {{c1::~~ ~~<b>safe edge</b>}} is a {{c2:: edge that is included in at <i>all</i> MSTs}}.	A {{c1::<b>safe edge</b>}} is an {{c2:: edge that is included in at <i>all</i> MSTs}}.

Tags: ETH::1._Semester::A&D::11._Minimum_Spanning_Trees

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: H{hvq(0Rc-

modified

Before

Front

Sei $G$ ein ungerichteter, gewichteter und zusammenhängender Graph.
Nehmen Sie an, dass es eine eindeutige Kante mit Gewicht $1$ gibt und, dass das Gewicht aller anderen Kanten strikt größer als $1$ ist.

Dann enthält jeder minimale Spannbaum von die Kante $e$.

Back

Dann enthält jeder minimale Spannbaum von die Kante $e$.

True

Wir wählen immer die Kante $e$, weil sie die günstigste Art die ZHK's zu verbinden ist.

Siehe Cut Property.

After

Front

Sei $G$ ein ungerichteter, gewichteter und zusammenhängender Graph.

Nehmen Sie an, dass es eine eindeutige Kante mit Gewicht $1$ gibt und, dass das Gewicht aller anderen Kanten strikt größer als $1$ ist.

Dann enthält jeder minimale Spannbaum von $G$ die Kante $e$.

Back

Sei $G$ ein ungerichteter, gewichteter und zusammenhängender Graph.

Nehmen Sie an, dass es eine eindeutige Kante mit Gewicht $1$ gibt und, dass das Gewicht aller anderen Kanten strikt größer als $1$ ist.

Dann enthält jeder minimale Spannbaum von $G$ die Kante $e$.

True

Wir wählen immer die Kante $e$, weil sie die günstigste Art die ZHK's zu verbinden ist.

Siehe Cut Property.

Field-by-field Comparison

Field	Before	After
Front	Sei $G$ ein ungerichteter, gewichteter und zusammenhängender Graph. <br>Nehmen Sie an, dass es eine eindeutige Kante mit Gewicht $1$ gibt und, dass das Gewicht aller anderen Kanten strikt größer als $1$ ist.<div>Dann enthält jeder minimale Spannbaum von die Kante $e$.</div>	Sei $G$ ein ungerichteter, gewichteter und zusammenhängender Graph. <br><br>Nehmen Sie an, dass es eine eindeutige Kante mit Gewicht $1$ gibt und, dass das Gewicht aller anderen Kanten strikt größer als $1$ ist.<br><br><div>Dann enthält jeder minimale Spannbaum von $G$ die Kante $e$.</div>

Tags: ETH::1._Semester::A&D::11._Minimum_Spanning_Trees

Note 3: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: Mv|.Tnx#vx

modified

Before

Front

A graph with distinct ege weights has one unique MST.

Back

A graph with distinct ege weights has one unique MST.

There is one unique safe-edge.

After

Front

A graph with distinct edge weights has one single unique MST.

Back

A graph with distinct edge weights has one single unique MST.

There is one unique safe-edge.

Field-by-field Comparison

Field	Before	After
Text	A graph with {{c1::distinct ege weights}} has {{c2::one unique MST}}.	A graph with {{c1::distinct edge weights}} has {{c2::one single unique MST}}.

Tags: ETH::1._Semester::A&D::11._Minimum_Spanning_Trees

Note 4: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: iHlSvEEQPk

modified

Before

Front

A graph $G$ is complete when it's set of edges is {{c2:: $\{\{u, v\} \ | \ u, v \in V, u \neq v\}$ }}.

Back

A graph $G$ is complete when it's set of edges is {{c2:: $\{\{u, v\} \ | \ u, v \in V, u \neq v\}$ }}.

After

Front

A graph $G$ is complete when it's set of edges is {{c2::$\{\{u, v\} \ | \ u, v \in V, u \neq v\}$ }}.

Back

A graph $G$ is complete when it's set of edges is {{c2::$\{\{u, v\} \ | \ u, v \in V, u \neq v\}$ }}.

Field-by-field Comparison

Field	Before	After
Text	A graph $G$ is {{c1::<b>complete</b>}} when it's set of edges is {{c2::~~ ~~$\{\{u, v\} \ \| \ u, v \in V, u \neq v\}$ }}.	A graph $G$ is {{c1::<b>complete</b>}} when it's set of edges is {{c2::$\{\{u, v\} \ \| \ u, v \in V, u \neq v\}$ }}.

Tags: ETH::1._Semester::A&D::07._Graphs::1._Introduction_to_Graphs

Note 5: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: BQ.+C9_=de

modified

Before

Front

The symbol $\perp$ denotes unsatisfiability.

Back

The symbol $\perp$ denotes unsatisfiability.

After

Front

The symbol $\perp$ denotes unsatisfiability.

Back

The symbol $\perp$ denotes unsatisfiability.

Field-by-field Comparison

Field	Before	After
Text	The symbol {{c1::~~ ~~$\perp$}} denotes {{c2:: unsatisfiability}}.	The symbol {{c1::$\perp$}} denotes {{c2:: unsatisfiability}}.

Tags: ETH::1._Semester::DiskMat::6._Logic::3._Elementary_General_Concepts_in_Logic::5._Satisfiability,_Tautology,_Consequence,_Equivalence

Note 6: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Ga3Xy8Kp9E

modified

Before

Front

The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}} (the union of their clause sets).

Back

The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}} (the union of their clause sets).

After

Front

The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is: {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}}

Back

The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is: {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}}

Field-by-field Comparison

Field	Before	After
Text	The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}} ~~(the {{c2::union of their clause sets}}).~~	The set of clauses associated with a set $M = \{F_1, \dots, F_k\}$ of formulas is: {{c1::\[\mathcal{K}(M) = \bigcup_{i=1}^k \mathcal{K}(F_i)\]}}

Tags: ETH::1._Semester::DiskMat::6._Logic::5._Propositional_Logic::5._The_Resolution_Calculus_for_Propositional_Logic

Note 7: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: H+<9!8uj.@

modified

Before

Front

The set of units of $R$ is denoted by $R^*$

Back

The set of units of $R$ is denoted by $R^*$

After

Front

The set of units of $R$ is denoted by $R^*$.

Back

The set of units of $R$ is denoted by $R^*$.

Field-by-field Comparison

Field	Before	After
Text	The set of units of $R$ is denoted by {{c1::$R^*$}}	The set of units of $R$ is denoted by {{c1::$R^*$}}.

Tags: ETH::1._Semester::DiskMat::5._Algebra::5._Rings_and_Fields::2._Units_and_the_Multiplicative_Group_of_a_Ring

Note 8: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Ic5Ww2Tp7G

modified

Before

Front

The set of clauses is the conjunction, it's only satisfied if every clause within is satisfied.

Back

The set of clauses is the conjunction, it's only satisfied if every clause within is satisfied.

After

Front

A set of clauses stands for the conjunction of the clauses, it's only satisfied if every clause within the set is satisfied.

Back

A set of clauses stands for the conjunction of the clauses, it's only satisfied if every clause within the set is satisfied.

Field-by-field Comparison

Field	Before	After
Text	~~The~~ set of clauses is the {{c1::<i>conjunction</i>}}, it's only satisfied if {{c2::every clause within is satisfied}}.	A set of clauses stands for the {{c1::<i>conjunction</i>}} of the clauses, it's only satisfied if {{c2::every clause within the set is satisfied}}.

Tags: ETH::1._Semester::DiskMat::6._Logic::5._Propositional_Logic::5._The_Resolution_Calculus_for_Propositional_Logic

Note 9: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Jd9Xy7Kn3H

modified

Before

Front

The empty clause $\emptyset$ (formula with no literals) corresponds to an unsatisfiable formula.

Back

The empty clause $\emptyset$ (formula with no literals) corresponds to an unsatisfiable formula.

After

Front

The empty clause $\emptyset$ (formula with no literals) corresponds to an unsatisfiable formula.

Back

The empty clause $\emptyset$ (formula with no literals) corresponds to an unsatisfiable formula.

A disjunction with no disjuncts is false.

Field-by-field Comparison

Field	Before	After
Extra		A disjunction with no disjuncts is false.

Tags: ETH::1._Semester::DiskMat::6._Logic::5._Propositional_Logic::5._The_Resolution_Calculus_for_Propositional_Logic

Note 10: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Ke6Zv4Rp8I

modified

Before

Front

The {{c1::empty clause set $\{\}$ (or $\emptyset$)}} corresponds to a tautology.

Back

The {{c1::empty clause set $\{\}$ (or $\emptyset$)}} corresponds to a tautology.

After

Front

The {{c1::empty clause set $\{\}$ (or $\emptyset$)}} corresponds to a tautology.

Back

The {{c1::empty clause set $\{\}$ (or $\emptyset$)}} corresponds to a tautology.

There are no clauses to satisfy.

Field-by-field Comparison

Field	Before	After
Extra		There are no clauses to satisfy.

Tags: ETH::1._Semester::DiskMat::6._Logic::5._Propositional_Logic::5._The_Resolution_Calculus_for_Propositional_Logic

Note 11: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Kq8Nx5Rm3J

modified

Before

Front

If $F$ is a tautology one also writes $\models F$. If it's unsatisfiable it can be written as $F \models \perp$.

Back

If $F$ is a tautology one also writes $\models F$. If it's unsatisfiable it can be written as $F \models \perp$.

After

Front

If $F$ is a tautology one also writes $\models F$. If is unsatisfiable it can be written as $F \models \perp$.

Back

If $F$ is a tautology one also writes $\models F$. If is unsatisfiable it can be written as $F \models \perp$.

Field-by-field Comparison

Field	Before	After
Text	If $F$ is a tautology one also writes {{c1::$\models F$}}. If it's unsatisfiable it can be written as {{c2::$F \models \perp$}}.	If $F$ is a tautology one also writes {{c1::$\models F$}}. If is unsatisfiable it can be written as {{c2::$F \models \perp$}}.

Tags: ETH::1._Semester::DiskMat::6._Logic::3._Elementary_General_Concepts_in_Logic::5._Satisfiability,_Tautology,_Consequence,_Equivalence

Note 12: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: Uo9Xv4Km8S

modified

Before

Front

For CNF construction, how do you form literals from a row?

Back

For CNF construction, how do you form literals from a row?

- If $A_i = 0$ in the row, take $A_i$
- If $A_i = 1$ in the row, take $\lnot A_i$

After

Front

For CNF construction, how do you form literals from a row in the truth table?

Back

For CNF construction, how do you form literals from a row in the truth table?

- If $A_i = 0$ in the row, take $A_i$
- If $A_i = 1$ in the row, take $\lnot A_i$

Field-by-field Comparison

Field	Before	After
Front	For CNF construction, how do you form literals from a row?	For CNF construction, how do you form literals from a row in the truth table?

Tags: ETH::1._Semester::DiskMat::6._Logic::5._Propositional_Logic::4._Normal_Forms

Note 13: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: lq:b}[Y<9t

modified

Before

Front

How is Lagrange interpolation for polynomials in a field defined?

Back

How is Lagrange interpolation for polynomials in a field defined?

Let $\beta_i = a(\alpha_i)$ for $i = 1, \dots, d+1$ where Then $\alpha_i$ distinct for all $i.$

$a(x)$ is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial $u_i(x)$ is: \[u_i(x) = \frac{(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}{(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}\]

Note that for $u_i(x)$ to be well-defined, all constant terms $\alpha_i - \alpha_j$ in the denominator must be invertible. This is guaranteed in a field since $\alpha_i - \alpha_j \neq 0$ for $i \neq j$ (as they are all distinct).

After

Front

How is Lagrange interpolation for polynomials in a field defined?

Back

How is Lagrange interpolation for polynomials in a field defined?

Let $\beta_i = a(\alpha_i)$ for $i = 1, \dots, d+1$ where $\alpha_i$ distinct for all $i.$

Field-by-field Comparison

Field	Before	After
Back	<p>Let $\beta_i = a(\alpha_i)$ for $i = 1, \dots, d+1$ where ~~Then~~ $\alpha_i$ distinct for all $i.$</p><p><br>$a(x)$ is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial $u_i(x)$ is: \[u_i(x) = \frac{(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}{(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}\]</p> <p>Note that for $u_i(x)$ to be well-defined, all constant terms $\alpha_i - \alpha_j$ in the denominator must be invertible. This is guaranteed in a field since $\alpha_i - \alpha_j \neq 0$ for $i \neq j$ (as they are all distinct).</p>	<p>Let $\beta_i = a(\alpha_i)$ for $i = 1, \dots, d+1$ where $\alpha_i$ distinct for all $i.$</p><p><br>$a(x)$ is given by Lagrange's Interpolation formula: \[a(x) = \sum_{i=1}^{d+1} \beta_i u_i(x)\] where the polynomial $u_i(x)$ is: \[u_i(x) = \frac{(x - \alpha_1) \cdots (x - \alpha_{i-1})(x - \alpha_{i+1}) \cdots (x - \alpha_{d+1})}{(\alpha_i - \alpha_1) \cdots (\alpha_i - \alpha_{i-1})(\alpha_i - \alpha_{i+1}) \cdots (\alpha_i - \alpha_{d+1})}\]</p> <p>Note that for $u_i(x)$ to be well-defined, all constant terms $\alpha_i - \alpha_j$ in the denominator must be invertible. This is guaranteed in a field since $\alpha_i - \alpha_j \neq 0$ for $i \neq j$ (as they are all distinct).</p>

Tags: ETH::1._Semester::DiskMat::5._Algebra::7._Polynomials_as_Functions::3._Polynomial_Interpolation PlsFix

Note 14: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: FTH7rOs5Fz

modified

Before

Front

Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}

Back

Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}

Fundamental theorem of algebra

After

Front

Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}.

Back

Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}.

Fundamental theorem of algebra

Field-by-field Comparison

Field	Before	After
Text	Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}	Every polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has {{c1:: a zero $\lambda \in \mathbb{C} $}}.

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 15: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: F]H_)uK%+@

modified

Before

Front

Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?

{{c1::Let $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$(matrix version of the recursion)}}
The eigenvalues $\lambda_1 = \frac{1+\sqrt{5{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5}}{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}
{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5}}$: $g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2$.}}
{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}

Back

Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?

{{c1::Let $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$(matrix version of the recursion)}}
The eigenvalues $\lambda_1 = \frac{1+\sqrt{5{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5}}{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}
{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5}}$: $g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2$.}}
{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}

After

Front

Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?

{{c1::Let $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} $(matrix version of the recursion)}}
{{c2::The eigenvalues $\lambda_1 = \frac{1+\sqrt{5} }{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5} }{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}
{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5} }$: $g_0 = \frac{1}{\sqrt{5} }v_1 - \frac{1}{\sqrt{5} }v_2$.}}
{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form: \[F_n = \frac{1}{\sqrt{5} } \left[\left(\frac{1+\sqrt{5} } {2}\right)^n - \left(\frac{1-\sqrt{5} } {2}\right)^n\right] \] }}

Back

Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?

{{c1::Let $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} $(matrix version of the recursion)}}
{{c2::The eigenvalues $\lambda_1 = \frac{1+\sqrt{5} }{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5} }{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}
{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5} }$: $g_0 = \frac{1}{\sqrt{5} }v_1 - \frac{1}{\sqrt{5} }v_2$.}}
{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form: \[F_n = \frac{1}{\sqrt{5} } \left[\left(\frac{1+\sqrt{5} } {2}\right)^n - \left(\frac{1-\sqrt{5} } {2}\right)^n\right] \] }}

Field-by-field Comparison

Field	Before	After
Text	Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?<br><ol><li>{{c1::Let  $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$(matrix version of the recursion)}}</li><li>{{c2::The eigenvalues $\lambda_1 = \frac{1+\sqrt{5}}{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5}}{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}</li><li>{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5}}$: $g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2$.}}</li><li>{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}</li></ol>	Express the Fibonacci formula $f_n = f_{n - 1} + f_{n - 2}$ as a matrix equation?<br><ol><li>{{c1::Let  $\mathbf{g}_{n+1} = M\mathbf{g}_n$, where $M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} $(matrix version of the recursion)}}</li><li>{{c2::The eigenvalues $\lambda_1 = \frac{1+\sqrt{5} }{2}$ (golden ratio $\phi$) and $\lambda_2 = \frac{1-\sqrt{5} }{2}$ are found, along with their eigenvectors $v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}$ . These eigenvectors are independent since $\lambda_1 \neq \lambda_2$ and thus they form a basis for $\mathbb{R}^2$.}}</li><li>{{c3::The initial state $\mathbf{g}_0$ is written as a linear combination of eigenvectors with coefficients $\pm\frac{1}{\sqrt{5} }$: $g_0 = \frac{1}{\sqrt{5} }v_1 - \frac{1}{\sqrt{5} }v_2$.}}</li><li>{{c4::Since $M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i$, we get the closed form: \[F_n = \frac{1}{\sqrt{5} } \left[\left(\frac{1+\sqrt{5} } {2}\right)^n - \left(\frac{1-\sqrt{5} } {2}\right)^n\right] \] }}</li></ol>

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1,9._Closed_Form_Recursion_(Fibonacci)

Note 16: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: PCBMoNL{vn

modified

Before

Front

The nullspace of $N(A) $ is equal to the nullspace of $N(A^\dagger)$. Proof Included

Back

The nullspace of $N(A) $ is equal to the nullspace of $N(A^\dagger)$. Proof Included

Intuitively this holds as $A^\dagger$ projects onto the $C(A)$.

Thus anything in $C(A)^\bot = N(A^\top)$ is projected to $0$. In other words, $\forall x \in C(A^\top)^\bot = N(A^\top)$ we have $A^\dagger x = 0$.
We conclude that $N(A) = C(A^\top)^\bot = N(A^\dagger)$.

After

Front

The nullspace of $N(A) $ is equal to the nullspace of $N(A^\dagger)$. Proof Included

Back

The nullspace of $N(A) $ is equal to the nullspace of $N(A^\dagger)$. Proof Included

Field-by-field Comparison

Field	Before	After
Text	The <b>nullspace of </b>$N(A) $ is equal to {{c1:: the nullspace of $N(A^\dagger)$}}. <i>Proof Included</i>	The <b>nullspace of </b>$N(A) $ is equal to {{c1:: the nullspace of $N(A^\dagger)$::Pseudoinverse}}. <i>Proof Included</i>

Tags: ETH::1._Semester::LinAlg::6._Applications_of_orthogonality_and_projections::4._The_Pseudoinverse,_also_known_as_Moore-Penrose_Inverse::2._Properties

Note 17: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: PtQN)*utrU

modified

Before

Front

Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent.

$W = V^\perp$
$\dim(V) + \dim(W) = n$
{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}

Back

Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent.

$W = V^\perp$
$\dim(V) + \dim(W) = n$
{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}

In words, this means that we can combine two orthogonal subspaces and create a new subspace, whose dimension is the sum of the two dimensions.

After

Front

Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent:

$W = V^\perp$
$\dim(V) + \dim(W) = n$
{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}

Back

Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent:

$W = V^\perp$
$\dim(V) + \dim(W) = n$
{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}

In words, this means that we can combine two orthogonal subspaces and create a new subspace, whose dimension is the sum of the two dimensions.

Field-by-field Comparison

Field	Before	After
Text	<div>Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent.</div><div><ol><li>{{c1::$W = V^\perp$}}</li><li>{{c2::$\dim(V) + \dim(W) = n$}}</li><li>{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}</li></ol></div><blockquote><ul> </ul></blockquote>	<div>Let $V, W$ be orthogonal subspaces of $\mathbb{R}^n$. Then the following statements are equivalent:</div><div><ol><li>{{c1::$W = V^\perp$}}</li><li>{{c2::$\dim(V) + \dim(W) = n$}}</li><li>{{c3::Every $u \in \mathbb{R}^n$ can be written as $u = v + w$ with unique vectors $v \in V$, $w \in W$}}</li></ol></div><blockquote><ul> </ul></blockquote>

Tags: ETH::1._Semester::LinAlg::5._Orthogonality_and_Projections::1._Orthogonality

Note 18: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: b$)8Q]2TlG

modified

Before

Front

$A$ is invertible if and only if there exists $B$ such that $AB = BA = I$

Back

$A$ is invertible if and only if there exists $B$ such that $AB = BA = I$

After

Front

$A$ is invertible if and only if there exists $B$ such that $AB = BA = I$,

Back

$A$ is invertible if and only if there exists $B$ such that $AB = BA = I$,

Field-by-field Comparison

Field	Before	After
Text	$A$ is invertible if and only if ~~{{c1::~~ there exists $B$ such that $AB = BA = I$~~ :: inverse condition equations~~ }}	$A$ is invertible if and only if there exists {{c1::$B$ such that $AB = BA = I$}},

Tags: ETH::1._Semester::LinAlg::2._Matrices::4._Invertible_and_Inverse_matrices::2._Definitions_and_basic_properties

Note 19: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: d[`]Ch#J!c

modified

Before

Front

Fundamental Theorem of Algebra?

Back

Fundamental Theorem of Algebra?

Any degree $n$ non-constant $n \geq 1$ polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has a (existance) zero: $\lambda \in \mathbb{C}$ such that $P(\lambda) = 0$.

After

Front

What is the fundamental theorem of algebra?

Back

What is the fundamental theorem of algebra?

Any degree $n$ polynomial $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0$ (with $n \geq 1$ and $a_n \neq 0$) has at least one zero $\lambda \in \mathbb{C}$ such that $P(\lambda) = 0$.

Field-by-field Comparison

Field	Before	After
Front	Fundamental Theorem of Algebra?	What is the fundamental theorem of algebra?
Back	<div>Any degree $n$ ~~non-constant $n \geq 1$~~ polynomial $P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0$ with $a_n \neq 0$ has a ~~(existance) zero: ~~$\lambda \in \mathbb{C}$ such that $P(\lambda) = 0$.</div>	<div>Any degree $n$ polynomial $P(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0$ (with $n \geq 1$ and $a_n \neq 0$) has at least one zero $\lambda \in \mathbb{C}$ such that $P(\lambda) = 0$.</div>

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 20: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: s!I_p%w(=W

modified

Before

Front

Any degree $n$ non constant ($n \geq 1$) polynomial $P(z)$ has {{c1:: $n$ zeros: $\lambda_1, \dots, \lambda_n \in \mathbb{C}$}}, perhaps with repetitions such that \[ P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n) \]

Back

After

Front

Any degree $n$ polynomial $P(z)$ (with $n \geq 1$) has {{c1::$n$ zeros $\lambda_1, \dots, \lambda_n \in \mathbb{C}$}}, possibly with repetitions, such that \[P(z) = a_n (z-\lambda_1)(z - \lambda_2) \cdots (z - \lambda_n)\]

Back

Field-by-field Comparison

Field	Before	After
Text	Any degree $n$ ~~non constant ($n \geq 1$) polynomial $P(z)\~~) has {{c1::~~ ~~\(n$ zeros: $\lambda_1, \dots, \lambda_n \in \mathbb{C}$}}, ~~perhaps {{c1::~~ with repetitions}} such that \[ P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n) \]~~<br>~~	Any degree $n$ polynomial $P(z)$ (with $n \geq 1$) has {{c1::$n$ zeros $\lambda_1, \dots, \lambda_n \in \mathbb{C}$}}, {{c1::possibly with repetitions}}, such that \[P(z) = a_n (z-\lambda_1)(z - \lambda_2) \cdots (z - \lambda_n)\]

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 21: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID:

uCC


            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1:: the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation :: inversions}}.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1:: the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation :: inversions}}.
    
    
    
    \((1, 3, 2)\) has one inversion.
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1::the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation::inversions}}.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1::the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation::inversions}}.
    
    
    
    \((1, 3, 2)\) has one inversion.

\(\text{sgn}(\sigma)=(−1)^k\) where \(k\) is the number of transpositions (swaps) needed to obtain \(σ\) from the identity.
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            The&nbsp;\(\text{sgn}(\sigma)\)&nbsp;where&nbsp;\(\sigma\)&nbsp;is a permutation is {{c1:: the parity of the number of elements that are out of order (inversions:&nbsp;\( i &lt; j \text{ and } \sigma(i) &gt; \sigma( j)\)) after applying the permutation :: inversions}}.
                            The&nbsp;\(\text{sgn}(\sigma)\)&nbsp;where&nbsp;\(\sigma\)&nbsp;is a permutation is {{c1::the parity of the number of elements that are out of order (inversions:&nbsp;\( i &lt; j \text{ and } \sigma(i) &gt; \sigma( j)\)) after applying the permutation::inversions}}.
                        
                        
                        
                            Extra
                            \((1, 3, 2)\)&nbsp;has one inversion.
                            \((1, 3, 2)\)&nbsp;has one inversion.<br><br>\(\text{sgn}(\sigma)=(−1)^k\)&nbsp;where&nbsp;\(k\)&nbsp;is the number of transpositions (swaps) needed to obtain&nbsp;\(σ\)&nbsp;from the identity.
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations

Field	Before	After
Text	The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1:: the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation :: inversions}}.	The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is {{c1::the parity of the number of elements that are out of order (inversions: \( i < j \text{ and } \sigma(i) > \sigma( j)\)) after applying the permutation::inversions}}.
Extra	\((1, 3, 2)\) has one inversion.	\((1, 3, 2)\) has one inversion.<br><br>\(\text{sgn}(\sigma)=(−1)^k\) where \(k\) is the number of transpositions (swaps) needed to obtain \(σ\) from the identity.


        
        
            Note 22: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Cloze

                GUID: uqAA$|?Lip
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors
  
  
    Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 \neq \lambda_2 \in \mathbb{R}\) be two distinct eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).
Then \(v_1\) and \(v_2\) are orthogonal. Proof Included
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors
  
  
    Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 \neq \lambda_2 \in \mathbb{R}\) be two distinct eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).
Then \(v_1\) and \(v_2\) are orthogonal. Proof Included
    
    
    
    \(\lambda_1 v_1 ^\top v_2 = (Av_1)^\top v_2\) \( = v_1^\top A ^\top v_2 = \) \(v_1^\top (Av_2)\) \( = \lambda_2 v_1^\top v_2\)
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors
  
  
    Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 \neq \lambda_2 \in \mathbb{R}\) two distinct eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).
Then \(v_1\) and \(v_2\) are orthogonal. Proof Included
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors
  
  
    Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 \neq \lambda_2 \in \mathbb{R}\) two distinct eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).
Then \(v_1\) and \(v_2\) are orthogonal. Proof Included
    
    
    
    \(\lambda_1 v_1 ^\top v_2 = (Av_1)^\top v_2\) \( = v_1^\top A ^\top v_2 = \) \(v_1^\top (Av_2)\) \( = \lambda_2 v_1^\top v_2\)
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            <div>Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 {{c2::\neq}} \lambda_2 \in \mathbb{R}\) be two {{c2::distinct}} eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).</div><div>Then \(v_1\) and \(v_2\)&nbsp;{{c1::are orthogonal}}.&nbsp;<i>Proof Included</i></div>
                            <div>Let \(A \in \mathbb{R}^{n \times n}\) be a symmetric matrix and \(\lambda_1 {{c2::\neq}} \lambda_2 \in \mathbb{R}\)&nbsp;two {{c2::distinct}} eigenvalues of \(A\) with corresponding eigenvectors \(v_1, v_2\).</div><div>Then \(v_1\) and \(v_2\)&nbsp;{{c1::are orthogonal}}.&nbsp;<i>Proof Included</i></div>
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors
                    
                    
                    
                
            
            
        
        
        
            Note 23: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Cloze

                GUID: wGNTPZMph;
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is  the parity of the number of row swaps necessary to get back to the identity .
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is  the parity of the number of row swaps necessary to get back to the identity .
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is  the parity of the number of row swaps necessary to get back to the identity .
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations
  
  
    The \(\text{sgn}(\sigma)\) where \(\sigma\) is a permutation is  the parity of the number of row swaps necessary to get back to the identity .
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            The&nbsp;\(\text{sgn}(\sigma)\)&nbsp;where&nbsp;\(\sigma\)&nbsp;is a permutation is {{c1:: the parity of the number of row swaps necessary to get back to the identity :: swaps}}.
                            The&nbsp;\(\text{sgn}(\sigma)\)&nbsp;where&nbsp;\(\sigma\)&nbsp;is a permutation is {{c1:: the parity of the number of row swaps necessary to get back to the identity ::swaps}}.
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::LinAlg::7._The_determinant::2._The_general_case::2._Permutations