Linearly dependent and independent rows

• Linear combination of rows
• Trivial linear combination of rows
• Non-trivial linear combination of rows
• Linear dependence of rows
• Linear independent of rows
• Examples

A linear combination of rows s₁, s₂, ..., s_l of matrix A is the following expression

α₁s₁ + α₂s₂ + ... + α_ls_l

A linear combination of rows is called trivial if all coefficients α_i are equal zero simultaneously.

A trivial linear combination of rows is equal to the zero row.

A linear combination of rows is called non-trivial if at least one of the coefficients α_i is not equal zero.

The system of rows is called linearly dependent, if there is a non-trivial linear combination of rows, which is equal to the zero row.

The system of rows is called linearly independent, if only trivial linear combination of rows are equal to the zero row (there is no non-trivial linear combination of rows equal to the zero row).

System of rows of square matrix are linearly independent if and only if the determinant of the matrix is ​​not equal to zero.

System of rows of square matrix are linearly dependent if and only if the determinant of the matrix is equals to zero.

Show that the system of rows {s₁ = {2   5}; s₂ = {4   10}} is linearly dependent.

Solution. Form a linear combination of these rows

α₁{2   5} + α₂{4   10}

Let us find for what values of α₁, α₂ the linear combination is equal to the zero row

α₁{2   5} + α₂{4   10} = {0   0}

This equation is equivalent to the following system of equations:

	2α1 + 4α2 = 0
	5α1 + 10α2 = 0

Divide the first equation by 2, and the second equation by 5:

	α1 + 2α2 = 0
	α1 + 2α2 = 0

The solution of this system may be any number α₁ and α₂ such that: α₁ = -2α₂, for example, α₂ = 1, α₁ = -2, and this means that the rows s₁ and s₂ are linearly dependent.

Show that the system of lines {s₁ = {2   5   1}; s₂ = {4   10   0}} is linearly independent.

Solution. Form a linear combination of these rows

α₁{2   5   1} + α₂{4   10   0}

Let us find for what values of α₁, α₂ the linear combination is equal to the zero row

α₁{2   5   0} + α₂{4   10   0} = {0   0   0}

This equation is equivalent to the following system of equations:

	2α1 + 4α2 = 0
	5α1 + 10α2 = 0
	α1 + 0α2 = 0

From the third equation gives α₁ = 0. Substituting this value in the first and second equation:

	2·0 + 4α2 = 0	=>		4α2 = 0	=>		α2 = 0
	5·0 + 10α2 = 0			10α2 = 0			α2 = 0
	α1 = 0			α1 = 0			α1 = 0

So as a linear combination of rows is equal to zero only when α₁ = 0 and α₂ = 0, the rows are linearly independent.