Laina fa'alagolago ma tuto'atasi: fa'amatalaga, fa'ata'ita'iga

In this publication, we will consider what a linear combination of strings is, linearly dependent and independent strings. We will also give examples for a better understanding of the theoretical material.

Defining a Linear Combination of Strings

Linear combination (LK) term s₁i₂, …, s_n Numera A called an expression of the following form:

αs₁ + αs₂ + … + αs_n

If all coefficients α_i are equal to zero, so LC is leai se taua. In other words, the trivial linear combination equals the zero row.

Faataitaiga: 0 · s₁ + 0 · s₂ + 0 · s₃

Accordingly, if at least one of the coefficients α_i is not equal to zero, then LC is non-trivial.

Faataitaiga: 0 · s₁ + 2 · s₂ + 0 · s₃

Linearly dependent and independent rows

The string system is fa'alagolago laina (LZ) if there is a non-trivial linear combination of them, which is equal to the zero line.

Hence it follows that a non-trivial LC can in some cases be equal to the zero string.

The string system is tuto'atasi laina (LNZ) if only the trivial LC is equal to the null string.

Faamatalaga:

• In a square matrix, the row system is an LZ only if the determinant of this matrix is ​​zero (le = 0).
• In a square matrix, the row system is an LIS only if the determinant of this matrix is ​​not equal to zero (le ≠ 0).

Faataitaiga o se faafitauli

Let’s find out if the string system is {s₁ = {3 4};s₂ = {9 12}} fa'alagolago laina.

Filifiliga:

1. First, let’s make a LC.

α₁{3 4} + a₂{9 12}.

2. Now let’s find out what values ​​should take α₁ и α₂so that the linear combination equals the null string.

α₁{3 4} + a₂{9 12} = {0 0}.

3. Let’s make a system of equations:

4. Divide the first equation by three, the second by four:

5. The solution of this system is any α₁ и α₂, With α₁ = -3a₂.

Mo se faʻataʻitaʻiga, afai α₂ = 2lea α₁ =-6. We substitute these values ​​into the system of equations above and get:

tali: so the lines s₁ и s₂ fa'alagolago laina.