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Γραμμική Ανεξαρτησία

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Γραμμική Ανεξαρτησία

Linear Independence



Είναι μία οικογένεια διανυσμάτων ενός Διανυσματικού Χώρου λέγονται "γραμμικά ανεξάρτητα" αν κανένα εξ αυτών δεν μπορεί να γραφεί ως Γραμμικός Συνδυασμός των άλλων.


ΕτυμολογίαEdit

Η ονομασία " " σχετίζεται ετυμολογικά με την λέξη " ".


ΓενικάEdit

A family of vectors which is not linearly independent is called linearly dependent. For instance, in the three-dimensional real vector space R3 we have the following example.


\begin{matrix}
\mbox{independent}\qquad\\
\underbrace{
  \overbrace{
    \begin{bmatrix}0\\0\\1\end{bmatrix},
    \begin{bmatrix}0\\2\\-2\end{bmatrix},
    \begin{bmatrix}1\\-2\\1\end{bmatrix}
  },
  \begin{bmatrix}4\\2\\3\end{bmatrix}
}\\
\mbox{dependent}\\
\end{matrix}


Here the first three vectors are linearly independent; but the fourth vector equals 9 times the first plus 5 times the second plus 4 times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; here we could just as well write the first vector as a linear combination of the last three.

\bold{v}_1 = \left(-\frac{5}{9}\right) \bold{v}_2 + \left(-\frac{4}{9}\right) \bold{v}_3 + \frac{1}{9} \bold{v}_4 .

Τυπικός ΟρισμόςEdit

A subset S of vector space V is called linearly dependent if there exist a finite number of distinct vectors v1, v2, ..., vn in S and scalars a1, a2, ..., an, not all zero, such that

 a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n = \mathbf{0}.

Note that the zero on the right is the zero vector, not the number zero.

If such scalars do not exist, then the vectors are said to be linearly independent. This condition can be reformulated as follows: Whenever a1, a2, ..., an are scalars such that

 a_1 \mathbf{v}_1 + a_2 \mathbf{v}_2 + \cdots + a_n \mathbf{v}_n = \mathbf{0},

we have ai = 0 for i = 1, 2, ..., n, i.e. only the trivial solution exists.

A set is linearly independent if and only if the only representations of the zero vector as linear combinations of its elements are trivial solutions.

More generally, let V be a vector space over a field K, and let {vi}iI be a family of elements of V. The family is linearly dependent over K if there exists a family {aj}jJ of elements of K, not all zero, such that

 \sum_{j \in J} a_j \mathbf{v}_j = \mathbf{0} \,

where the index set J is a nonempty, finite subset of I.

A set X of elements of V is linearly independent if the corresponding family {x}xX is linearly independent.

Equivalently, a family is dependent if a member is in the linear span of the rest of the family, i.e., a member is a linear combination of the rest of the family.

A set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.

Γεωμετρική ΣημασίαEdit

A geographic example may help to clarify the concept of linear independence. A person describing the location of a certain place might say, "It is 5 miles north and 6 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered a 2-dimensional vector space (ignoring altitude). The person might add, "The place is 7.81 miles northeast of here." Although this last statement is true, it is not necessary.

In this example the "5 miles north" vector and the "6 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "7.81 miles northeast" vector is a linear combination of the other two vectors, and it makes the set of vectors linearly dependent, that is, one of the three vectors is unnecessary.

Note that in this example, any of the three vectors may be described as a linear combination of the other two. While it might be inconvenient, one could describe "6 miles east" in terms of north and northeast. (For example, "Go 5 miles south (mathematically, −5 miles north) and then go 7.81 miles northeast.") Similarly, the north vector is a linear combination of the east and northeast vectors.

Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, n linearly independent vectors are required to describe any location in n-dimensional space.

Παράδειγμα IEdit

The vectors (1, 1) and (−3, 2) in R2 are linearly independent.

Απόδειξη Edit

Let λ1 and λ2 be two real numbers such that

 (1, 1) \lambda_1 + (-3, 2) \lambda_2 = (0, 0) . \,\!

Taking each coordinate alone, this means

 \begin{align}
 \lambda_1 - 3 \lambda_2 &{}= 0 , \\
 \lambda_1 + 2 \lambda_2 &{}= 0 . 
\end{align}

Solving for λ1 and λ2, we find that λ1 = 0 and λ2 = 0.

Εναλλακτική Μέθοδος με χρήση Οριζουσών Edit

An alternative method uses the fact that n vectors in Rn are linearly dependent if and only if the determinant of the matrix formed by taking the vectors as its columns is zero.

In this case, the matrix formed by the vectors is

A = \begin{bmatrix}1&-3\\1&2\end{bmatrix} . \,\!

We may write a linear combination of the columns as

 A \Lambda = \begin{bmatrix}1&-3\\1&2\end{bmatrix} \begin{bmatrix}\lambda_1 \\ \lambda_2 \end{bmatrix} . \,\!

We are interested in whether AΛ = 0 for some nonzero vector Λ. This depends on the determinant of A, which is

 \det A = 1\cdot2 - 1\cdot(-3) = 5 \ne 0 . \,\!

Since the determinant is non-zero, the vectors (1, 1) and (−3, 2) are linearly independent.

When the number of vectors equals the dimension of the vectors, the matrix is square and hence the determinant is defined.

Otherwise, suppose we have m vectors of n coordinates, with m < n. Then A is an n×m matrix and Λ is a column vector with m entries, and we are again interested in AΛ = 0. As we saw previously, this is equivalent to a list of n equations. Consider the first m rows of A, the first m equations; any solution of the full list of equations must also be true of the reduced list. In fact, if 〈i1,…,im〉 is any list of m rows, then the equation must be true for those rows.

 A_{{\lang i_1,\dots,i_m} \rang} \Lambda = \bold{0} . \,\!

Furthermore, the reverse is true. That is, we can test whether the m vectors are linearly dependent by testing whether

 \det A_{{\lang i_1,\dots,i_m} \rang} = 0 \,\!

for all possible lists of m rows. (In case m = n, this requires only one determinant, as above. If m > n, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.

Παράδειγμα IIEdit

Let V = Rn and consider the following elements in V:

\begin{matrix}
\mathbf{e}_1 & = & (1,0,0,\ldots,0) \\
\mathbf{e}_2 & = & (0,1,0,\ldots,0) \\
& \vdots \\
\mathbf{e}_n & = & (0,0,0,\ldots,1).\end{matrix}

Then e1, e2, ..., en are linearly independent.

Απόδειξη Edit

Suppose that a1, a2, ..., an are elements of R such that

 a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + \cdots + a_n \mathbf{e}_n = 0 . \,\!

Since

 a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + \cdots + a_n \mathbf{e}_n = (a_1 ,a_2 ,\ldots, a_n) , \,\!

then ai = 0 for all i in {1, ..., n}.

Παράδειγμα III Edit

Let V be the vector space of all functions of a real variable t. Then the functions et and e2t in V are linearly independent.

Απόδειξη Edit

Suppose a and b are two real numbers such that

aet + be2t = 0

for all values of t. We need to show that a = 0 and b = 0. In order to do this, we divide through by et (which is never zero) and subtract to obtain

bet = −a

In other words, the function bet must be independent of t, which only occurs when b = 0. It follows that a is also zero.

Παράδειγμα IVEdit

The following vectors in R4 are linearly dependent.


\begin{matrix}
\\
    \begin{bmatrix}1\\4\\2\\-3\end{bmatrix},
    \begin{bmatrix}7\\10\\-4\\-1\end{bmatrix} \mathrm{and},
    \begin{bmatrix}-2\\1\\5\\-4\end{bmatrix}
\\
\\
\end{matrix}

Απόδειξη Edit

We need to find scalars \lambda_1, \lambda_2 and \lambda_3 such that


\begin{matrix}
\\
\lambda_1  \begin{bmatrix}1\\4\\2\\-3\end{bmatrix}+
\lambda_2  \begin{bmatrix}7\\10\\-4\\-1\end{bmatrix}+
\lambda_3  \begin{bmatrix}-2\\1\\5\\-4\end{bmatrix}=
           \begin{bmatrix}0\\0\\0\\0\end{bmatrix}
\end{matrix}

Forming the simultaneous equations:


\begin{align}
  \lambda_1& \;+  7\lambda_2& &- 2\lambda_3& = 0\\
 4\lambda_1& \;+ 10\lambda_2& &+  \lambda_3& = 0\\
 2\lambda_1& \;-  4\lambda_2& &+ 5\lambda_3& = 0\\
-3\lambda_1& \;-   \lambda_2& &- 4\lambda_3& = 0\\
\end{align}

we can solve (using for example Gaussian elimination) to obtain:


\begin{align}
  \lambda_1 &= -3/2  \\
  \lambda_2 &= 1/2  \\
  \lambda_3 &= 1  \\
\end{align}

Since these are nontrivial results, the vectors are linearly dependent.


Ο Προβολικός Χώρος της Γραμμικής ΕξάρτησηςEdit

A linear dependence among vectors v1, ..., vn is a tuple (a1, ..., an) with n scalar components, not all zero, such that

a_1 \mathbf{v}_1 + \cdots + a_n \mathbf{v}_n=0. \,

If such a linear dependence exists, then the n vectors are linearly dependent. It makes sense to identify two linear dependences if one arises as a non-zero multiple of the other, because in this case the two describe the same linear relationship among the vectors. Under this identification, the set of all linear dependences among v1, ...., vn is a projective space.


Εσωτερική ΑρθρογραφίαEdit

ΒιβλιογραφίαEdit

ΙστογραφίαEdit

  • [ ]


Ikl.jpg Κίνδυνοι ΧρήσηςIkl.jpg

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.



Επίσης,
Οι διάφοροι "Εξωτερικοί Σύνδεσμοι (Links)"
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν

IonnKorr-System-00-goog.png



>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)


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