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Definition. A linear combination of rows s_{1}, s_{2}, ..., s_{l} of matrix A is the following expression
α_{1}s_{1} + α_{2}s_{2} + ... + α_{l}s_{l} Definition. A linear combination of rows is called trivial if all coefficients α_{i} are equal zero simultaneously.Note. A trivial linear combination of rows is equal to the zero row.Definition. A linear combination of rows is called Definition. The system of rows is called linearly dependent, if there is a nontrivial linear combination of rows, which is equal to the zero row.Definition. The system of rows is called linearly independent, if only trivial linear combination of rows are equal to the zero row (there is no nontrivial linear combination of rows equal to the zero row).Note. System of rows of square matrix are linearly independent if and only if the determinant of the matrix is not equal to zero.Note. System of rows of square matrix are linearly dependent if and only if the determinant of the matrix is equals to zero.Example 1. Show that the system of rows {s_{1} = {2 5}; s_{2} = {4 10}} is linearly dependent.
Solution. Form a linear combination of these rows α_{1}{2 5} + α_{2}{4 10} Let us find for what values of α_{1}, α_{2} the linear combination is equal to the zero row α_{1}{2 5} + α_{2}{4 10} = {0 0} This equation is equivalent to the following system of equations:
Divide the first equation by 2, and the second equation by 5:
The solution of this system may be any number α_{1} and α_{2} such that: α_{1} = 2α_{2}, for example, α_{2} = 1, α_{1} = 2, and this means that the rows s_{1} and s_{2} are linearly dependent. Example 2. Show that the system of lines {s_{1} = {2 5 1}; s_{2} = {4 10 0}} is linearly independent.
Solution. Form a linear combination of these rows α_{1}{2 5 1} + α_{2}{4 10 0} Let us find for what values of α_{1}, α_{2} the linear combination is equal to the zero row α_{1}{2 5 0} + α_{2}{4 10 0} = {0 0 0} This equation is equivalent to the following system of equations:
From the third equation gives α_{1} = 0. Substituting this value in the first and second equation:
So as a linear combination of rows is equal to zero only when α_{1} = 0 and α_{2} = 0, the rows are linearly independent. MatrixMatrix Definition. Main informationSystem of linear equations  matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix Add the comment 