Anki Deck Changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: A8P;P^G,v4

modified

Before

Front

Runtime of sorting an array containing only \(1, 0\)?

Back

Runtime of sorting an array containing only \(1, 0\)?

Using bucketsort, we can achieve \(O(n)\). this seems contradictory.
We go through the array once, counting occurences of \(0\) as x. We then add \(x\) zeros in the beginning and fill the rest with 1s.

After

Front

Runtime of sorting an array containing only \(1, 0\)?

Back

Runtime of sorting an array containing only \(1, 0\)?

Using bucketsort, we can achieve \(O(n)\).

We go through the array once, counting occurences of \(0\) as x. We then add \(x\) zeros in the beginning and fill the rest with 1s.

Field-by-field Comparison

Field	Before	After
Back	Using bucketsort, we can achieve \(O(n)\). ~~this seems contradictory.~~<br>We go through the array once, counting occurences of \(0\) as x. We then add \(x\) zeros in the beginning and fill the rest with 1s.	Using bucketsort, we can achieve \(O(n)\). <br><br>We go through the array once, counting occurences of \(0\) as x. We then add \(x\) zeros in the beginning and fill the rest with 1s.

Tags: ETH::1._Semester::A&D::04._Sorting_Algorithms

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Algorithms
GUID: Ah4U@kYYNJ

modified

Before

Front

Runtime of Linear Search?

Back

Runtime of Linear Search?

\(\Theta(n)\) as we go through the entire list once.

After

Front

Runtime of Linear Search?

Back

Runtime of Linear Search?

\(\Theta(n)\) as we go through the entire list once.

Field-by-field Comparison

Field	Before	After
Approach	~~Linear search simply goes~~ through the entire list and compares the current element to the one we are ~~searching~~.	Go through the entire list and compare the current element to the one we are looking for.

Tags: ETH::1._Semester::A&D::03._Searching_Algorithms::1._Linear_Search

Note 3: ETH::A&D

Deck: ETH::A&D
Note Type: Algorithms
GUID: DD]dBe%Sn|

modified

Before

Front

Runtime of Binary Search?

Back

Runtime of Binary Search?

\(O(\log(n))\) (optimal)

After

Front

Runtime of Binary Search?

Back

Runtime of Binary Search?

\(O(\log(n))\) (optimal)

Field-by-field Comparison

Field	Before	After
Approach	~~You s~~tart in the middle ~~and if the middle element is not the one you're searching, you~~ recur~~se o~~n the ~~left OR right side (depending on the middle elements size)~~.	Start in the middle of the array. <br><br>If the middle element is the target element, return the current index.<br><br>Else if the middle elment is larger (smaller) than the target element, continue recursively on the left (right) half of the array.

Tags: ETH::1._Semester::A&D::03._Searching_Algorithms::2._Binary_Search

Note 4: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: M,?u9cw(S%

modified

Before

Front

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O notation)

Back

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O notation)

After

Front

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O-notation)

Back

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) \(O(n \log(n))\) (O-notation)

Field-by-field Comparison

Field	Before	After
Text	{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}}  \(\leq\) {{c2::\(O(n \log(n))\)}} (O notation)	{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}}  \(\leq\) {{c2::\(O(n \log(n))\) (O-notation)}}

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

Note 5: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Cloze
GUID: yg-tkTB|,7

modified

Before

Front

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\)}} (Sum)

Back

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\)}} (Sum)

After

Front

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\) (Sum)}}

Back

{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}} \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\) (Sum)}}

Field-by-field Comparison

Field	Before	After
Text	{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}}  \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\)~~}} (Sum)~~	{{c1:: \(\sum_{i = 1}^{n} i\log(i)\)}}  \(\leq\) {{c2::\(\sum_{i = 1}^n n \log(n) = n^2 \log n\) (Sum)}}

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

Note 6: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: AluZ0L@#]a

modified

Before

Front

What does it mean for a function \(f: A \to B\) to be injective (one-to-one)?

Back

What does it mean for a function \(f: A \to B\) to be injective (one-to-one)?

For \(a \neq a'\) we have \(f(a) \neq f(a')\). No two distinct values are mapped to the same function value (no "collisions").

After

Front

What does it mean for a function \(f: A \to B\) to be injective?

Back

What does it mean for a function \(f: A \to B\) to be injective?

For \(a \neq a'\) we have \(f(a) \neq f(a')\).

No two distinct values are mapped to the same function value (no "collisions"). This is also called "one-to-one".

Field-by-field Comparison

Field	Before	After
Front	What does it mean for a function \(f: A \to B\) to be injective ~~(one-to-one)~~?	What does it mean for a function \(f: A \to B\) to be injective?
Back	For \(a \neq a'\) we have \(f(a) \neq f(a')\). No two distinct values are mapped to the same function value (no "collisions").	For \(a \neq a'\) we have \(f(a) \neq f(a')\). <br><br>No two distinct values are mapped to the same function value (no "collisions"). This is also called "one-to-one".

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::DUPLICATE

Note 7: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Gw~6}3;R1[

modified

Before

Front

\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).

Back

\((A;\preceq)\) is a poset. If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).

After

Front

Consider the poset \((A;\preceq)\).

If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).

Back

Consider the poset \((A;\preceq)\).

If \(\{a,b\}\) have a greatest lower bound, then it is called the meet of \(a\) and \(b\) (also denoted \(a \land b\)).

Field-by-field Comparison

Field	Before	After
Text	\((A;\preceq)\) ~~is a poset.~~ If \(\{a,b\}\) have a {{c2::greatest lower bound}}, then it is called the {{c1::<b>meet </b>of \(a\) and \(b\) (also denoted \(a \land b\)).}}<br>	Consider the poset \((A;\preceq)\). <br><br>If \(\{a,b\}\) have a {{c2::greatest lower bound}}, then it is called the {{c1::<b>meet </b>of \(a\) and \(b\) (also denoted \(a \land b\)).}}<br>

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices

Note 8: ETH::DiskMat

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

HTIS


            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
  
  
    What are the two trivial equivalence relations on a set \(A\)?

1.  Complete relation \(A \times A\) → single equivalence class \(A\)
2. {{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
  
  
    What are the two trivial equivalence relations on a set \(A\)?

1.  Complete relation \(A \times A\) → single equivalence class \(A\)
2. {{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
  
  
    What are the two trivial equivalence relations on a set \(A\)?
 Complete relation \(A \times A\) → single equivalence class \(A\)
{{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::4._Equivalence_Relations::1._Definition_of_Equivalence_Relations
  
  
    What are the two trivial equivalence relations on a set \(A\)?
 Complete relation \(A \times A\) → single equivalence class \(A\)
{{c2:: Identity relation → equivalence classes are all singletons \(\{a\}\)}}
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
                            After
                        
                    
                    
                        
                        
                            Text
                            What are the two trivial equivalence relations on a set \(A\)?<br><br>1. {{c1::&nbsp;<strong>Complete relation</strong>&nbsp;\(A \times A\)&nbsp;→ single equivalence class&nbsp;\(A\)}}<br>2.&nbsp;{{c2::&nbsp;<strong>Identity relation</strong>&nbsp;→ equivalence classes are all singletons&nbsp;\(\{a\}\)}}
                            What are the two trivial equivalence relations on a set \(A\)?<br><ol><li>{{c1::&nbsp;<strong>Complete relation</strong>&nbsp;\(A \times A\)&nbsp;→ single equivalence class&nbsp;\(A\)}}</li><li>{{c2::&nbsp;<strong>Identity relation</strong>&nbsp;→ equivalence classes are all singletons&nbsp;\(\{a\}\)}}</li></ol>
                        
                        
                    
                
            
            

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

Field	Before	After
Text	What are the two trivial equivalence relations on a set \(A\)?<br><~~br>1.~~ {{c1:: <strong>Complete relation</strong> \(A \times A\) → single equivalence class \(A\)}}<~~br>2. ~~{{c2:: <strong>Identity relation</strong> → equivalence classes are all singletons \(\{a\}\)}}	What are the two trivial equivalence relations on a set \(A\)?<br><ol><li>{{c1:: <strong>Complete relation</strong> \(A \times A\) → single equivalence class \(A\)}}</li><li>{{c2:: <strong>Identity relation</strong> → equivalence classes are all singletons \(\{a\}\)}}</li></ol>


        
        
            Note 9: ETH::DiskMat
            
                Deck: ETH::DiskMat

                Note Type: Horvath Cloze

                GUID: otWm4$@-u8
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices
  
  
    \((A;\preceq)\) is a poset. If \(\{a,b\}\) have a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices
  
  
    \((A;\preceq)\) is a poset. If \(\{a,b\}\) have a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices
  
  
    Consider the poset \((A;\preceq)\). 

If \(\{a,b\}\) has a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices
  
  
    Consider the poset \((A;\preceq)\). 

If \(\{a,b\}\) has a least upper bound, then it is called the join of \(a\) and \(b\) (also denoted \(a \lor b\)).
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
                            After
                        
                    
                    
                        
                        
                            Text
                            \((A;\preceq)\) is a poset. If&nbsp;\(\{a,b\}\) have a {{c2::least upper bound}}, then it is called the {{c1::<b>join&nbsp;</b>of&nbsp;\(a\) and&nbsp;\(b\) (also denoted&nbsp;\(a \lor b\)).}}
                            Consider the poset&nbsp;\((A;\preceq)\). <br><br>If&nbsp;\(\{a,b\}\) has a {{c2::least upper bound}}, then it is called the {{c1::<b>join&nbsp;</b>of&nbsp;\(a\) and&nbsp;\(b\) (also denoted&nbsp;\(a \lor b\)).}}
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::5._Partial_Order_Relations::5._Meet,_Join,_and_Lattices
                    
                    
                    
                
            
            
        
        
        
            Note 10: ETH::DiskMat
            
                Deck: ETH::DiskMat

                Note Type: Horvath Cloze

                GUID: v
            

            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
  
  
    What two properties must a relation \(f: A \to B\) have to be a function?

1.  Totally defined: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 
2.  Well-defined: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
  
  
    What two properties must a relation \(f: A \to B\) have to be a function?

1.  Totally defined: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 
2.  Well-defined: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
    
    
  



                        
                    
                    
                
                
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                        Front
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::DUPLICATE
  
  
    What two properties must a relation \(f: A \to B\) have to be a function?
 Total-definedness: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 
 Well-definedness: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions PlsFix::DUPLICATE
  
  
    What two properties must a relation \(f: A \to B\) have to be a function?
 Total-definedness: \(\forall a \in A \ \exists b \in B : a \ f \ b\) 
 Well-definedness: \(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            What two properties must a relation \(f: A \to B\) have to be a function?<br><br>1. {{c1::&nbsp;<strong>Totally defined</strong>:&nbsp;\(\forall a \in A \ \exists b \in B : a \ f \ b\)&nbsp;}}<br>2.&nbsp;{{c2::&nbsp;<strong>Well-defined</strong>:&nbsp;\(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)}}
                            What two properties must a relation \(f: A \to B\) have to be a function?<br><ol><li>{{c1::&nbsp;<strong>Total-definedness</strong>:&nbsp;\(\forall a \in A \ \exists b \in B : a \ f \ b\)&nbsp;}}</li><li>{{c2::&nbsp;<strong>Well-definedness</strong>:&nbsp;\(\forall a \in A \ \forall b, b' \in B : (a \ f \ b \land a \ f \ b' \rightarrow b = b')\)}}</li></ol>
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions
                    
                    
                    
                    PlsFix::DUPLICATE
                    
                
            
            
        

        
        
            Note 11: ETH::EProg
            
                Deck: ETH::EProg

                Note Type: Horvath Cloze

                GUID: m%x4[`&]1%
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::EProg::3._Control_Structures::3._Methods
  
  
    Only  name and input types  determine the signature of a method in Java.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::EProg::3._Control_Structures::3._Methods
  
  
    Only  name and input types  determine the signature of a method in Java.
    
    
  



                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::EProg::3._Control_Structures::3._Methods
  
  
    Only  names and input types  determine the signature of a method in Java.
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::EProg::3._Control_Structures::3._Methods
  
  
    Only  names and input types  determine the signature of a method in Java.
    
    
  



                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Text
                            Only {{c1:: name and input types }} determine the signature of a method in Java.
                            Only {{c1:: names and input types }} determine the signature of a method in Java.
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::EProg::3._Control_Structures::3._Methods
                    
                    
                    
                
            
            
        
        
        
            Note 12: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Classic

                GUID: Qp5sodd?T?
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
  
  
    What is the definition of a linear transformation or a linear functional?
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
  
  
    What is the definition of a linear transformation or a linear functional?
    
    a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\) is called a linear transformation or a linear functional if the linearity axiom holds for it 

linearity axiom: \(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
  






                        
                    
                    
                
                
                    After
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
  
  
    When is a function considered to be a linear transformation or a linear functional?
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
  
  
    When is a function considered to be a linear transformation or a linear functional?
    
    If the linearity axiom holds for it:

\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
  






                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
                            Before
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                            Front
                            What is the definition of a linear transformation or a linear functional?
                            When is a function considered to be a linear transformation or a linear functional?
                        
                        
                        
                            Back
                            a function&nbsp;\(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\)&nbsp;is called a linear transformation or a linear functional if the&nbsp;<b>linearity axiom</b>&nbsp;holds for it&nbsp;<br><br><b>linearity axiom:&nbsp;</b>\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
                            If the&nbsp;<b>linearity axiom</b>&nbsp;holds for it:<br><b><br></b>\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::LinAlg::2._Matrices::2._Matrices_and_linear_transformations::1._Matrix_transformations
                    
                    
                    
                
            
            
        
        
        
            Note 13: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Classic

                GUID: f>Z/u`5f-r
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What does \(N(A) = \{0\}\) mean?
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What does \(N(A) = \{0\}\) mean?
    
    it means that all the columns of the matrix are independent
  






                        
                    
                    
                
                
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                        Front
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What does \(N(A) = \{0\}\) mean?
  



                        
                    
                    
                        Back
                        
                            
  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What does \(N(A) = \{0\}\) mean?
    
    That all the columns of the matrix are independent.
  






                        
                    
                    
                
            
            

            
            
            
                Field-by-field Comparison
                
                    
                        
                            Field
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                            it means that all the columns of the matrix are independent
                            That all the columns of the matrix are independent.
                        
                        
                    
                
            
            

            
            
            
                Tags:
                
                    
                    
                    
                    
                    ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
                    
                    
                    
                
            
            
        
        
        
            Note 14: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Cloze

                GUID: nkL&a6|Q;d
            
            modified
            

            
            
            
                
                    Before
                    
                    
                        Front
                        
                            
  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::3._Span_of_vectors PlsFix::ClozeThatBish
  
  
    The span of m linearly independent vectors is \({{c1::\mathbb{R}^m}}\).
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::3._Span_of_vectors PlsFix::ClozeThatBish
  
  
    The span of m linearly independent vectors is \({{c1::\mathbb{R}^m}}\).
    
    
    
    This also means that a matrix in \(\mathbb{R}^{n \times n}\) with rank(A) = n spans all of the space.
  



                        
                    
                    
                
                
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::3._Span_of_vectors PlsFix::ClozeThatBish
  
  
    The span of m linearly independent vectors is \({{c1::\mathbb{R}^m}}\).
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::3._Span_of_vectors PlsFix::ClozeThatBish
  
  
    The span of m linearly independent vectors is \({{c1::\mathbb{R}^m}}\).
    
    
    
    This also means that a matrix in \(\mathbb{R}^{n \times n}\) with rank(A) = n spans the entire space.
  



                        
                    
                    
                
            
            

            
            
            
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                            This also means that a matrix in&nbsp;\(\mathbb{R}^{n \times n}\)&nbsp;with rank(A) = n spans all of the space.
                            This also means that a matrix in&nbsp;\(\mathbb{R}^{n \times n}\)&nbsp;with rank(A) = n spans the entire space.
                        
                        
                    
                
            
            

            
            
            
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                    ETH::1._Semester::LinAlg::1._Vectors::3._Linear_(in)dependence::3._Span_of_vectors
                    
                    PlsFix::ClozeThatBish
                    
                    
                    
                
            
            
        
        
        
            Note 15: ETH::LinAlg
            
                Deck: ETH::LinAlg

                Note Type: Horvath Classic

                GUID: uWwT2a*Vb[
            
            modified
            

            
            
            
                
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  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What is the nullspace of a matrix?  
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What is the nullspace of a matrix?  
    
    all vectors that when multiplied by the matrix give the 0-vector out
\(N(A) := \{x\in \mathbb{R} : Ax = \boldsymbol{0} \}\)
  






                        
                    
                    
                
                
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  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What is the nullspace of a matrix?  
  



                        
                    
                    
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  ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace
  
  
    What is the nullspace of a matrix?  
    
    The set of vectors that give the 0-vector when multiplied with the given matrix.

\(N(A) := \{x\in \mathbb{R} : Ax = \boldsymbol{0} \}\)
  






                        
                    
                    
                
            
            

            
            
            
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                            all vectors that when multiplied by the matrix give the 0-vector out<br>\(N(A) := \{x\in \mathbb{R} : Ax = \boldsymbol{0} \}\)
                            The set of vectors that give the 0-vector&nbsp;when multiplied with the given matrix.<br><br>\(N(A) := \{x\in \mathbb{R} : Ax = \boldsymbol{0} \}\)
                        
                        
                    
                
            
            

            
            
            
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                    ETH::1._Semester::LinAlg::2._Matrices::1._Matrices_and_linear_combinations::5._Nullspace

Field	Before	After
Front	What is ~~the definition of~~ a linear transformation or a linear functional?	When is a function considered to be a linear transformation or a linear functional?
Back	~~a function \(T: \mathbb{R}^n \rightarrow \mathbb{R}^m / \ T: \mathbb{R}^n \rightarrow \mathbb{R}\) is called a linear transformation or a linear functional i~~f the <b>linearity axiom</b> holds for it~~ ~~<br><br><b>~~linearity axiom: ~~</b>\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)	If the <b>linearity axiom</b> holds for it:<br><b><br></b>\(T(\lambda_1x_1 + \lambda_2x_2) = \lambda_1T(x_1) + \lambda_2T(x_2)\)