Linear dependent and independent rows: definition, examples

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, …, s_n matrix A called an expression of the following form:

αs₁ + αs₂ + … + αs_n

If all coefficients α_i 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 α_i 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 {s₁ = {3 4};s₂ = {9 12}} linearly dependent.

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.