Anki Deck Changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: D{#5DM:PFn

modified

Before

Front

$O(1) \leq$ (Name the next bigger function)

Back

$O(1) \leq$ (Name the next bigger function)

$\leq O(\log(n))$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(1) \leq O(\log(n))$

Back

Choose a tight bound!

$O(1) \leq O(\log(n))$

Field-by-field Comparison

Field	Before	After
Text	$O(1) \leq~~$ <i>(Name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(1)}} \leq {{c2::O(\log(n))}}$
Extra	~~$\leq O(\log(n))$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: G9Xwk@MZZb

modified

Before

Front

If $f \leq O(h)$

Back

If $f \leq O(h)$

$\forall C > 0$ we have $c \cdot f \leq O(h)$

After

Front

What does $f \leq O(h)$ mean exactly?

Back

What does $f \leq O(h)$ mean exactly?

$\forall C > 0$ we have $c \cdot f \leq O(h)$

Field-by-field Comparison

Field	Before	After
Front	If $f \leq O(h)$	What does $f \leq O(h)$ mean exactly?

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation

Note 3: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: KgqO:Z_iT+

modified

Before

Front

$O(\log(n)) \leq$ (name the next bigger function)

Back

$O(\log(n)) \leq$ (name the next bigger function)

$\leq O(n)$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(\log(n)) \leq O(n)$

Back

Choose a tight bound!

$O(\log(n)) \leq O(n)$

Field-by-field Comparison

Field	Before	After
Text	$O(\log(n)) \leq~~$ <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(\log(n))}} \leq {{c2::O(n)}}$
Extra	~~$\leq O(n)$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 4: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: NGdgt3D+pd

modified

Before

Front

We can ignore the base of a logarithm only if it's not in the exponent.

Back

We can ignore the base of a logarithm only if it's not in the exponent.

$e^{\log_2 n} \neq \Theta(e^{\log_3 n})$ as $e^{\log_2 n - \log_3 n} = e^{\ln n (\frac{1}{\ln(2)} - \frac{1}{\ln(3)})}$ goes to $\infty$

After

Front

We can ignore the base of a logarithm only if it's not in the exponent.

Back

We can ignore the base of a logarithm only if it's not in the exponent.

$e^{\log_2 n} \neq \Theta(e^{\log_3 n})$ as $e^{\log_2 n - \log_3 n} = e^{\ln n (\frac{1}{\ln(2)} - \frac{1}{\ln(3)})}$ goes to $\infty$

Field-by-field Comparison

Field	Before	After
Text	We can ignore the base of a logarithm ~~{{c1::~~ only if it's not in the exponent}}.	We can ignore the base of a logarithm only if {{c1::it's not in the exponent}}.

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation

Note 5: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: QA~(,/7jXV

modified

Before

Front

$O(k^n) \leq$ (name the next bigger function)

Back

$O(k^n) \leq$ (name the next bigger function)

$\leq O(n!)$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(k^n) \leq O(n!)$

Back

Choose a tight bound!

$O(k^n) \leq O(n!)$

Field-by-field Comparison

Field	Before	After
Text	$O(k^n) \leq~~$ <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(k^n)}} \leq {{c2::O(n!)}}$
Extra	~~$\leq O(n!)$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 6: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: fa/9a*g+D.

modified

Before

Front

How can I get the lower bound on the function $n!$ ?

Back

How can I get the lower bound on the function $n!$ ?

I can only take for example the largest 90% of elements $n! \geq 1 \cdot 2 \cdot ... \cdot n \geq n/10 \cdot ... \cdot n$

$\geq (n/10)^{0.9n}$

After

Front

How can one get a lower bound for the function $n!$ ?

Back

How can one get a lower bound for the function $n!$ ?

One could simply take only the largest 90% of elements: $n! \geq 1 \cdot 2 \cdot ... \cdot n \geq n/10 \cdot ... \cdot n$

$\geq (n/10)^{0.9n}$

Field-by-field Comparison

Field	Before	After
Front	How can I get ~~the~~ lower bound on the function $n!$ ?	How can one get a lower bound for the function $n!$ ?
Back	~~I can only take for example~~ the largest 90% of elements $n! \geq 1 \cdot 2 \cdot ... \cdot n \geq n/10 \cdot ... \cdot n$<div>$\geq (n/10)^{0.9n}$</div>	One could simply take only the largest 90% of elements: $n! \geq 1 \cdot 2 \cdot ... \cdot n \geq n/10 \cdot ... \cdot n$<div>$\geq (n/10)^{0.9n}$</div>

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation

Note 7: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: k#~pL>w{_$

modified

Before

Front

What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?

{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are differentiable}}
{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they tend to infinity)}}
{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ never equals to 0}}

Back

What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?

{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are differentiable}}
{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they tend to infinity)}}
{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ never equals to 0}}

This guarantees that we can take the fraction f/g.

After

Front

What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?

{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are differentiable}}
{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they tend to infinity)}}
{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ never equals 0}}

Back

What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?

{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are differentiable}}
{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they tend to infinity)}}
{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ never equals 0}}

This guarantees that we can take the fraction f/g.

Field-by-field Comparison

Field	Before	After
Text	What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?<br><ol><li>{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are <b>differentiable</b>}}<br></li><li>{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they <b>tend to infinity)</b>}}<br></li><li>{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ <b>never equals to 0</b>}}</li></ol>	What are the prerequisites for $f$ and $g$ to apply l'Hôpital's?<br><ol><li>{{c1::$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ are <b>differentiable</b>}}<br></li><li>{{c2::$\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = \infty$ (they <b>tend to infinity)</b>}}<br></li><li>{{c3::$\forall x \in \mathbb{R}^{+} , g'(x) \neq 0$ and $g$ <b>never equals 0</b>}}</li></ol>

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation

Note 8: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: ms|2]oAD;M

modified

Before

Front

If $T(n) = aT(n/ 2) + Cn^b$, then the O-Notation gives

Back

If $T(n) = aT(n/ 2) + Cn^b$, then the O-Notation gives

$T(n) = \Theta(...)$

After

Front

If $T(n) = aT(n/ 2) + Cn^b$, then we get which type of O-Notation?

Back

If $T(n) = aT(n/ 2) + Cn^b$, then we get which type of O-Notation?

$T(n) = \Theta(...)$

Field-by-field Comparison

Field	Before	After
Front	If  $T(n) = aT(n/ 2) + Cn^b$, then ~~the~~ O-Notation ~~gives~~	If  $T(n) = aT(n/ 2) + Cn^b$, then we get which type of O-Notation?

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation PlsFix::ClozeThatBish

Note 9: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: sap]uaOK!q

modified

Before

Front

$O(n^k) \leq$ (name the next bigger function)

Back

$O(n^k) \leq$ (name the next bigger function)

$\leq O(k^n)$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(n^k) \leq O(k^n)$

Back

Choose a tight bound!

$O(n^k) \leq O(k^n)$

Field-by-field Comparison

Field	Before	After
Text	$O(n^k) \leq~~$ <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(n^k)}} \leq {{c2::O(k^n)}}$
Extra	~~$\leq O(k^n)$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 10: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: uxhT]f%32k

modified

Before

Front

Master Theorem: If $b = \log_2(a)$ then {{c2:: $T(n) \leq O(n^{\log_2 a} \cdot \log n)$}}.

Back

Master Theorem: If $b = \log_2(a)$ then {{c2:: $T(n) \leq O(n^{\log_2 a} \cdot \log n)$}}.

The recursive and non-recursive work is balanced.

After

Front

Master Theorem: If $b = \log_2(a)$ then {{c2:: $T(n) \leq O(n^{\log_2 a(=b)} \cdot \log n)$}}.

Back

Master Theorem: If $b = \log_2(a)$ then {{c2:: $T(n) \leq O(n^{\log_2 a(=b)} \cdot \log n)$}}.

The recursive and non-recursive work is balanced.

Field-by-field Comparison

Field	Before	After
Text	Master Theorem: If {{c1:: $b = \log_2(a)$}} then {{c2:: $T(n) \leq O(n^{\log_2 a} \cdot \log n)$}}.	Master Theorem: If {{c1:: $b = \log_2(a)$}} then {{c2:: $T(n) \leq O(n^{\log_2 a(=b)} \cdot \log n)$}}.

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation

Note 11: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: yOa^MOZU`,

modified

Before

Front

$O(n) \leq$ (name the next bigger function)

Back

$O(n) \leq$ (name the next bigger function)

$\leq O(n \log(n))$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(n) \leq O(n \log(n))$

Back

Choose a tight bound!

$O(n) \leq O(n \log(n))$

Field-by-field Comparison

Field	Before	After
Text	$O(n) \leq~~$ <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(n)}} \leq {{c2::O(n \log(n))}}$
Extra	~~$\leq O(n \log(n))$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 12: ETH::A&D

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

modified

Before

Front

$O(n \log(n)) \leq$ (name the next bigger function)

Back

$O(n \log(n)) \leq$ (name the next bigger function)

$\leq O(n^k)$ (name the next smaller function)

After

Front

Choose a tight bound!

$O(n \log(n)) \leq O(n^k)$

Back

Choose a tight bound!

$O(n \log(n)) \leq O(n^k)$

Field-by-field Comparison

Field	Before	After
Text	$O(n \log(n)) \leq~~$  <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>${{c1::O(n \log(n))}} \leq {{c2::O(n^k)}}$
Extra	~~$\leq O(n^k)$ <i>(name the next smaller function)</i>~~

Tags: ETH::1._Semester::A&D::02._Asymptotic_Notation::3._O-Notation::Ordering

Note 13: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: A0$~M#^-(C

modified

Before

Front

If $F \models G$ in predicate logic, what can we conclude about validity?

Back

If $F \models G$ in predicate logic, what can we conclude about validity?

If $F$ is valid, then $G$ is also valid. (Logical consequence preserves validity)

After

Front

If $F \models G$ in predicate logic, what can we conclude via validity?

Back

If $F \models G$ in predicate logic, what can we conclude via validity?

If $F$ is valid, then $G$ is also valid. (Logical consequence preserves validity)

Field-by-field Comparison

Field	Before	After
Front	If $F \models G$ in predicate logic, what can we conclude ~~about~~ validity?	If $F \models G$ in predicate logic, what can we conclude via validity?

Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::5._Logical_Formulas_vs._Mathematical_Statements::2._Mathematical_Statements_about_Formulas

Note 14: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: BCPARdin7?

modified

Before

Front

What is the logical rule for case distinction?

Back

What is the logical rule for case distinction?

For every $k$ we have: \[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\]
(If at least one case occurs, and all cases imply $B$, then $B$ holds)

After

Front

What is the logical principle behind case distinction?

Back

What is the logical principle behind case distinction?

For every $k$ we have: \[(A_1 \lor \dots \lor A_k) \land (A_1 \rightarrow B) \land \dots \land (A_k \rightarrow B) \models B\]
(If at least one case occurs, and all cases imply $B$, then $B$ holds)

Field-by-field Comparison

Field	Before	After
Front	What is the logical ~~rule for~~ case distinction?	What is the logical principle behind case distinction?

Tags: ETH::1._Semester::DiskMat::2._Math._Reasoning,_Proofs,_and_a_First_Approach_to_Logic::6._Some_Proof_Patterns::05._Case_Distinction

Note 15: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: C;65zxNGcG

modified

Before

Front

The inverse relation is {{c1::$ a \ \rho \ b \iff b \ \hat{\rho} \ a$}}.

Back

The inverse relation is {{c1::$ a \ \rho \ b \iff b \ \hat{\rho} \ a$}}.

Example: Inverse of parent relation is childhood relation. Also written as $ \rho^{-1}$

After

Front

The definition of inverse relation is $ a \ \rho \ b \iff{{c1:: b \ \hat{\rho} \ a}}$.

Back

The definition of inverse relation is $ a \ \rho \ b \iff{{c1:: b \ \hat{\rho} \ a}}$.

Example: Inverse of parent relation is childhood relation. Also written as $ \rho^{-1}$

Field-by-field Comparison

Field	Before	After
Text	The {{c2::inverse relation}} is ~~{{c1::~~$ a \ \rho \ b \iff b \ \hat{\rho} \ a$}}.	The definition of {{c2::inverse relation}} is $ a \ \rho \ b \iff{{c1:: b \ \hat{\rho} \ a}}$.

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::4._The_Inverse_of_a_Relation

Note 16: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: K,;}YIg:-h

modified

Before

Front

A relation $\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is a subset of $A\times B$. If $A = B$, then $\rho$ is called a relation on $A$.

Back

A relation $\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is a subset of $A\times B$. If $A = B$, then $\rho$ is called a relation on $A$.

After

Front

A relation $\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is a subset of $A\times B$.

If $A = B$, then $\rho$ is called a relation on $A$.

Back

A relation $\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is a subset of $A\times B$.

If $A = B$, then $\rho$ is called a relation on $A$.

Field-by-field Comparison

Field	Before	After
Text	A <b>relation </b>$\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is {{c1::a subset of~~ ~~$A\times B$.}} If $A = B$, then $\rho$ is called {{c1::a <i>relation on</i> $A$.}}	A <b>relation </b>$\rho$ from a set $A$ to a set $B$ (also called an $(A,B)$-relation) is a {{c1::subset}} of {{c1::$A\times B$.}} <br><br>If $A = B$, then $\rho$ is called {{c1::a <i>relation on</i> $A$.}}

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::1._The_Relation_Concept

Note 17: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: tmXe!J(6@%

modified

Before

Front

Is antisymmetric the negation of symmetric?

Back

Is antisymmetric the negation of symmetric?

NO! Antisymmetry is NOT the negation of symmetry. They are independent properties.

After

Front

Is antisymmetric the negation of symmetric?

Back

Is antisymmetric the negation of symmetric?

NO! Antisymmetry is NOT the negation of symmetry. They are independent properties.

A relation is both symmetric and antisymmetric if and only if it's a subset of the identity relation (i.e., $R \subseteq \{(a, a) : a \in A\}$). This includes the empty relation $R = \emptyset$ as a degenerate case.

Field-by-field Comparison

Field	Before	After
Back	<strong>NO!</strong> Antisymmetry is NOT the negation of symmetry. They are independent properties.	<strong>NO!</strong> Antisymmetry is NOT the negation of symmetry. They are independent properties.<br><br>A relation is both symmetric and antisymmetric if and only if it's a subset of the identity relation (i.e., $R \subseteq \{(a, a) : a \in A\}$). This includes the empty relation $R = \emptyset$ as a degenerate case.

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::3._Relations::6._Special_Properties_of_Relations

Note 18: ETH::EProg

Deck: ETH::EProg
Note Type: Horvath Cloze
GUID: z3M|&J1r.r

modified

Before

Front

Not every EBNF language (Sprache) can be described with repetition (Wiederholung).

Back

Not every EBNF language (Sprache) can be described with repetition (Wiederholung).

After

Front

Not every EBNF language (Sprache) can be described just with repetition (Wiederholung).

Back

Not every EBNF language (Sprache) can be described just with repetition (Wiederholung).

Field-by-field Comparison

Field	Before	After
Text	Not every EBNF language (Sprache) can be described with {{c2:: repetition (Wiederholung)}}.	Not every EBNF language (Sprache) can be described just with{{c2:: repetition (Wiederholung)}}.

Tags: ETH::1._Semester::EProg::1._EBNF::6._Recursion

Note 19: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: M527x=(6av

modified

Before

Front

The euclidian norm of $\textbf{v}$ is the number

Back

The euclidian norm of $\textbf{v}$ is the number

$|| \textbf{v} || := \sqrt{\textbf{v} \cdot \textbf{v}}$

This is also called the 2-norm.

After

Front

The euclidian norm of $\textbf{v}$ is defined as?

Back

The euclidian norm of $\textbf{v}$ is defined as?

$|| \textbf{v} || := \sqrt{\textbf{v} \cdot \textbf{v}}$

This is also called the 2-norm.

Field-by-field Comparison

Field	Before	After
Front	The euclidian norm of $\textbf{v}$ is ~~the number ~~	The euclidian norm of $\textbf{v}$ is defined as?

Tags: ETH::1._Semester::LinAlg::1._Vectors::2._Scalar_products,_lengths_and_angles::2._Euclidean_norm

Note 20: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: eY$X2~xJ5/

modified

Before

Front

Give the three definitions for linear independence:

None of the vectors is a linear combination of the other ones.
{{c2::There are no scalars $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$. $\mathbf{0}$ can only be written as a trivial combination of the vectors.}}
None of the vectors is a linear combination of the previous ones.

Back

Give the three definitions for linear independence:

None of the vectors is a linear combination of the other ones.
{{c2::There are no scalars $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$. $\mathbf{0}$ can only be written as a trivial combination of the vectors.}}
None of the vectors is a linear combination of the previous ones.

After

Front

Give the three definitions for linear independence:

None of the vectors is a linear combination of the other ones.
{{c2::There are no scalars $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$. ($\mathbf{0}$ can only be written as a trivial combination of the vectors.)}}
None of the vectors is a linear combination of the previous ones.

Back

Give the three definitions for linear independence:

None of the vectors is a linear combination of the other ones.
{{c2::There are no scalars $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$. ($\mathbf{0}$ can only be written as a trivial combination of the vectors.)}}
None of the vectors is a linear combination of the previous ones.

Field-by-field Comparison

Field	Before	After
Text	Give the three definitions for linear independence:<br><ol><li>{{c1::None of the vectors is a linear combination of the other ones.}}</li><li>{{c2::There are no scalars  $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$.~~ ~~$\mathbf{0}$ can only be written as a trivial combination of the vectors.}}<br></li><li>{{c3::None of the vectors is a linear combination of the previous ones.}}</li></ol>	Give the three definitions for linear independence:<br><ol><li>{{c1::None of the vectors is a linear combination of the other ones.}}</li><li>{{c2::There are no scalars  $\lambda_1, ..., \lambda_n$ besides 0, 0, ..., 0 such that $\sum_{i = 1}^n \lambda_i v_i = \mathbf{0}$. ($\mathbf{0}$ can only be written as a trivial combination of the vectors.)}}<br></li><li>{{c3::None of the vectors is a linear combination of the previous ones.}}</li></ol>

Tags: ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions

Note 21: ETH::LinAlg

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

l[e7/3<


            modified
            

            
            
            
                
                    Before
                    
                    
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions
  
  
    If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly independent and \(A\lambda = A\mu = x\) are two ways of writing vector x as a linear combination of the vectors v then:
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions
  
  
    If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly independent and \(A\lambda = A\mu = x\) are two ways of writing vector x as a linear combination of the vectors v then:
    
    \(\lambda \ \text{and} \ \mu\) are the same vector.
Linear combinations are unique if all vectors are independent.
  






                        
                    
                    
                
                
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions PlsFix::ClozeThatBish
  
  
    If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly independent and \(A\lambda = A\mu = x\) are two ways of writing vector x as a linear combination of the vectors v then:
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions PlsFix::ClozeThatBish
  
  
    If columns \(v_1, v_2, ..., v_n\) of \(A\) are linearly independent and \(A\lambda = A\mu = x\) are two ways of writing vector x as a linear combination of the vectors v then:
    
    \(\lambda \ \text{and} \ \mu\) are the exact same vector of coefficients.
Linear combinations are unique if all vectors are independent.
  






                        
                    
                    
                
            
            

            
            
            
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                            \(\lambda \ \text{and} \ \mu\)&nbsp;are the same vector.<div><br></div><div>Linear combinations are unique if all vectors are independent.</div>
                            \(\lambda \ \text{and} \ \mu\)&nbsp;are the exact same vector of coefficients.<div><br></div><div>Linear combinations are unique if all vectors are independent.</div>
                        
                        
                    
                
            
            

            
            
            
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                    ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::2._Alternative_definitions
                    
                    
                    
                    PlsFix::ClozeThatBish

Field	Before	After
Text	\(O(1) \leq~~\) <i>(Name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(1)}} \leq {{c2::O(\log(n))}}\)
Extra	~~\(\leq O(\log(n))\) <i>(name the next smaller function)</i>~~

Field	Before	After
Text	\(O(\log(n)) \leq~~\) <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(\log(n))}} \leq {{c2::O(n)}}\)
Extra	~~\(\leq O(n)\) <i>(name the next smaller function)</i>~~

Field	Before	After
Text	\(O(k^n) \leq~~\) <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(k^n)}} \leq {{c2::O(n!)}}\)
Extra	~~\(\leq O(n!)\) <i>(name the next smaller function)</i>~~

Field	Before	After
Text	\(O(n^k) \leq~~\) <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(n^k)}} \leq {{c2::O(k^n)}}\)
Extra	~~\(\leq O(k^n)\) <i>(name the next smaller function)</i>~~

Field	Before	After
Text	\(O(n) \leq~~\) <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(n)}} \leq {{c2::O(n \log(n))}}\)
Extra	~~\(\leq O(n \log(n))\) <i>(name the next smaller function)</i>~~

Field	Before	After
Text	\(O(n \log(n)) \leq~~\)  <i>(name the next bigger function)</i>~~	Choose a tight bound!<br><br>\({{c1::O(n \log(n))}} \leq {{c2::O(n^k)}}\)
Extra	~~\(\leq O(n^k)\) <i>(name the next smaller function)</i>~~