Contents
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 s1With2, …, sn matrix A called an expression of the following form:
αs1 + αs2 + … + αsn
If all coefficients αi are equal to zero, so LC is trivial. In other words, the trivial linear combination equals the zero row.
For example: 0 · s1 + 0 · s2 + 0 · s3
Accordingly, if at least one of the coefficients αi is not equal to zero, then LC is non-trivial.
For example: 0 · s1 + 2 · s2 + 0 · s3
Linearly dependent and independent rows
The string system is linearly dependent (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 linearly independent (LNZ) if only the trivial LC is equal to the null string.
Notes:
- In a square matrix, the row system is an LZ only if the determinant of this matrix is zero (the = 0).
- In a square matrix, the row system is an LIS only if the determinant of this matrix is not equal to zero (the ≠ 0).
Example of a problem
Let’s find out if the string system is
Decision:
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.
For example, if α2 = 2then α1 =-6. We substitute these values into the system of equations above and get:
Answer: so the lines s1 и s2 linearly dependent.