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Mathematics > Linear Algebra > Flashcards

Linear Algebra Flashcards

1
Q

What is the definizion of a vector space ?

A

A vector space V is a non empty set of vectors on which are defined two operazions (addition and multiplication) by scalars, subject to 10 axioms that must hold.

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2
Q

What are the 10 axioms of the vector space?

A

1) i due vettori sommati continuano a far parte del vector space
2) vale la proprietà commutativa nella somma dei due vettori
3) vale la proprietà distributiva nella somma di tre vettori
4) esiste un vettore 0 elemento del vector space tale che v1+0 = v1
5) esiste ed appartiene al vector space un vettore z = -v1 (negativo di v1)
6) un vettore moltiplicato per uno scalare continua a far parte del vector space
7) moltiplicando uno scalare per la somma di due vettori vale la proprietà distributiva
8) moltiplicando un vettore per la somma di due scalari vale la proprietà distributiva
9) moltiplicando il prodotto tra uno scalare e il vettore per un altro scalare vale la proprietà distributiva
10) moltiplicando il vettore per 1 otteniamo sempre il vettore stesso

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3
Q

Quali sono i due modi per sommare due vettori?

A

GRAFICAMENTE = diagonale del parallelogramma
ALGEBRICAMENTE = sommo la prima riga e sotto la seconda riga scrivendo i vettori in verticale

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4
Q

Come capire il verso del vettore in base allo scalare per cui è moltiplicato?

A

Alfa > 0 , alfa×v = v, stessa direzione
Alfa < 0, alfa×v e v hanno direzioni opposte
Alfa = 0, alfa×v = 0

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5
Q

Whats the definition of a subspace ?

A

A subspace H of a vector is a subset of V that satisfies these 3 statements:
1) zero vector elemento del subspace
2) H is closed under vector addition (la somma di due vettori appartenuti ad H è ancora parte di H)
3) H is closed under multiplication of a vector by a scalar (la moltiplicazione di un vettore appartenente ad H per uno scalare è ancora parte di H)

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6
Q

Può essere H = R^2 \ {00} subspace of R^2?

A

No perché non soddisfa la prima proprietà dei subspace

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7
Q

Cosa si intende per span?

A

È il set di tutti i vettori che possono essere linearmente combinati

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8
Q

Cosa sono i generatori ? Quanti ne devono essere in numero ?

A

Sono I vettori numerici dello span

Ex: H = span {generators}

Il numero dei generatori deve essere maggiore o uguale della dimensione del vector space

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9
Q

Quando i vettori si dicono linearmente indipendenti ?

A

av1 + bv2 + … + dvn = 0, con tutti gli scalari uguali a 0

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10
Q

qual è la definizione di Base?

A

let H be a subspace of a vector space V, a set of vectors in V is a basis for H if
1) H = span {v1,..,vn} with v1,..,vn are generators for H
2) {v1,…,vn} is a linearly independent set

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11
Q

what does “dimension of H” mean?

A

dimension of H = number of vectors in a basis of H

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12
Q

how to find Generators in a span where vectors are expressed through letters?

A

example: in a vector defined in R^3, express the rule respect to a letter and then substitute in order to have only 2 letters left. The numerical part of the two letters are the two generators.

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13
Q

what to do when you’re asked to find LI vectors but you find out instead that they’re LD?

A

you have to remove a vector

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14
Q

How to prove that span {v1, v2, v3} = span {v1, v2}?

A

scrivo tutti i vettori del primo span, v1, v2 e v3 e vedo che uno dei tre è LD per gli altri due quindi il primo span è sottoinsieme del secondo span

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15
Q

what does the spanning set theorem state?

A

let {v1.. vn} set of vectors in V and H=span {v1.. vn}

1) if one of the vectors (for example Vn) is a linear combination of {v1.. vn} then {v1.. vn-1} still generates H.
span {v1.. vn} = span {v1.. vn-1}

2) if H is not 0 there exists a basis for H

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16
Q

If there are more than N (N= dimension) vectors in a vector space are they LD or LI?

A

Any set of vectors belonging to V containing more than N vectors is a set of LD vectors

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17
Q

what values should have H in its dimension in order to be a subspace of V?

A

dim H minore o uguale a dim V per essere un subspace

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18
Q

H0w to computer the DOT PRODUCT?

A

Remark: the result is a scalar!!

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19
Q

What are the 4 properties of the dot product?

A
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20
Q

Whats the length or norm of a vector?

A
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21
Q

Whats the property regarding the norm between a scalar and a vector?

A

||c × v|| = |c| × ||v||

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22
Q

Whats a unit vector? How can you build it?

A

||v|| = 1

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23
Q

How to measure the distante between 2 vectors?

A
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24
Q

What happening when the dot product between two vectors is equal to 0?

A

They’re called orthogonal vectors

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25
Q

Whats the orthogonal complement?

A

W⊥ is the set of vectors that are orthogonal to all the vectors of W, so W⊥ is said to be the orthogonal complement of W

26
Q

How to find W⊥ ?

A

You have to prove the rule of the dot product between two vectors equal to 0

27
Q

Come mi aiuta questa formula se devo determinare se ivettori sono LD o LI?

dimW + dimW⊥ = N

A

EX: dimW =1
N = 3
quindi dimW⊥ = 2
Se lo span conteneva 2 generatori, allora sono per forza LI
(N>dimW allora sono LD)

28
Q

Whats the definition of a matrix?

A
29
Q

Whats a square matrix?

A

A matrix where m=n (rows=columns)

30
Q

Whats a diagonal matrix?

A

A particular square matrix where all around the diagonal there are 0 elements

31
Q

Whats a 0 matrix?

A

Matrix whose entries are all equal to 0

32
Q

How does the matrix define the vector space?

A

The space of matrices with real coefficients m and n is a vector space

33
Q

How to computer the sum between matrices?

A
34
Q

What are the conditions to compute the product between two matrices? And how to compute it?

A

CONDITION: n of A must be equal to m of B (A first matrix and B second matrix)
HOW TO COMPUTE: vedi foto

35
Q

What is an identity matrix?

A
36
Q

Whats a Transpose matrix?

A

A^T is n×m whise columns are formed from the corresponding rows of A

37
Q

What are the properties of the transpose?

A

(AB)^T diverso da A^T × B^T

38
Q

How to compute the determinant of a 2×2 matrix?

A

Det A = ad - bc

39
Q

How to compute the determinant of big square matrices? N>2

A

Remark: choose line with more 0 entries

40
Q

Whats an inverse matrix?

A
41
Q

What are 3 properties of the inverse matrix?

A

1) A^-1 is invertible=> (A^-1)^-1 = A
2) (AB)^-1 = B^-1 × A^-1
3) (A^T)^-1 = (A^-1)^T

42
Q

How to determine the inverse of a 2×2 matrix?

A
43
Q

How to find W⊥?

A

1) compute a basis for W
2) find W⊥ using the definition imposing that the scalar product between a general vector of R^n and any vector of the basis (computed at step 1) is equal to 0

44
Q

What is the value of W⊥ × W (scalar product)?

A

Is equal to 0

45
Q

Whats the relation between determinant and linearly independent vectors?

A

1) m>n , LD
2) m = n (square matrix), LI if and only if the determinant is different from 0. If the determinant is equal to 0 they are LD
3) m<n, icannot compute the determinant so I will use the rank. To determine if they are LD or LI, use the definition of LI, look at the vectors and do the determinant (=0 means they are LD)

46
Q

If the determinant of a matrix is equal to k, how can we enstabilish if the vectors are LD or LI?

A

If k=0 they are LD, if not they are LI

47
Q

Whats the definition of cofactor?

A

The cofactor is the determinant of the matrix obtained by eliminating the row and column in the matrix that contains the element and then multiplying by +1 or -1 according to the position of the element

48
Q

Whats the rank of a matrix?

A
49
Q

What are the steps of computing rank of A?

A

1) start checking the max possible value for rank(A), which is min{m,n}=: rmax
If rank(A) = rmax there exists a rmax×rmax submatrix of A having determinant different from 0
(I have to check all rmax×rmax subatrices until I find the one with det different from 0, if they are all 0 it means rank(A) minor or qual to rmax-1
2) check value of rmax - 1 if the rmax is not the one.
3) check photo below

50
Q

When do we use the det(A) and the rank(A)?

A

Rank is for all matrices m>n, even the square matrices.
The det is only for square matrices

51
Q

Whats the Sarrous Rule?

A

Its another way to compute the determinant of a square matrix (ONLY 3×3 matrices)

52
Q

Tell me what are the 2 conditions and what the rank theorem states

A

DEF 1: rank(A) is the dimension of Col(A)
DEF 2: the nullity of a matrix A, written Nullity(A), is the dimension of Nul(A)

rank(A) = dim Col(A) (number of columns)
rank(A) = dim Nul(A) (number of free variables)

“if A is a matrix with n columns, then rank(A) + nullity(A) = n”

53
Q

Whats the matrix form used to describe a system of linear equations?

A

AX = b, where A (m×n) is the coefficient matrix, X (n×1) is the vector of variables and b (m×1) is the vector of coefficients.

When b=0, AX=0, the system is homogeneous

54
Q

When a system is inconsistent ?

A

When it does not have any solution

55
Q

What does the Rouche Capelli theorem state?

A

The system Ax = b (x € R^n) is consistent if and only if rank(A) = rank(A|b) = r. The system has inf^(n-r) solutions.

If r=n, 1 solution
If r</= n-1, inf^(n-r)

PS: if rank(A) is different from rank(A|b) the system is inconsistent

56
Q

What does the exponent “n-r” mean in the rouche capelli theorem?

A

“n-r” are the free variables, i.e. variables that cannot express the non free variables (static) or, equivalently, that cannot be expressed as function of the static variables

57
Q

What are the steps to solve a linear equation system?

A

1) divide rank(A) from b
2) write the rank(A|b)
3) determine the rank(A) from the max possible value (det(A) different from 0)
4) determine tha rank (A|b) from the max possible value (det(A|b) different from 0)
4) by watching at the matrix (m×n) if rank(A) = rank(A|b)
5) see with Rouche Capelli theorem if the system is consistent and how many solutions it admits [inf^(n-r)]

58
Q

How to decide what are the free and the static variables?

A

The static variables are the columns of the square matrix with det different from 0 used to compute the rank. The others are free variables

59
Q

Come risolvere i sistemi di eq lineari con Cramer ?

A

-eliminare dal sistema di equazioni eventuali equazioni linearmente dipendenti e fare in modo che il numero di equazioni sia uguale al numero di incognite
-scrivere il sistema con le variabili libere a destra (come se fossero la “b”, termini noti)
-trasformare nella forma Ax = b (A coefficienti delle variabili fisse, x variabili fisse, b variabili libere)
-calcolare ad una ad una le variabili fisse:
Xi = 1/detA × det (…), con detA diverso da 0
Nel determinante copio le colonne così come stanno nella matrice completa, ma mettendo la colonna di “b” al posto della colonna “i’

60
Q

Cosa è la nullità di una matrice e come si calcola?

A

La nullità di A è la dimensione dello spazio delle soluzioni del sistema linear3 omogeneo per cui Ax=0

La calcolo facendo: numero incognite - rango matrice dei coefficienti

Mathematics (3 decks)

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