A matrix is a rectangular array of numbers, symbols, or expressions organized into rows and columns. It is commonly used in mathematics and other scientific fields to represent and solve systems of equations.
An M×n matrix with m linearly independent rows is a matrix with m rows and n columns where each row is linearly independent. This means that no row can be expressed as a linear combination of the other rows.
Having linearly independent rows in a matrix is important because it allows for unique solutions to be found when solving systems of equations. It also helps to simplify calculations and makes it easier to determine the rank and determinant of a matrix.
To determine if a matrix has linearly independent rows, you can use the reduced row-echelon form (RREF) method. If the RREF of the matrix has a pivot in every column, then the rows are linearly independent. Additionally, you can also use the determinant of the matrix to determine if the rows are linearly independent, as a non-zero determinant indicates linear independence.
The significance of having m linearly independent rows in an M×n matrix is that it allows for a maximum of m independent equations to be solved. This means that the matrix has a unique solution or a solution space with m dimensions.