Define the row space of a matrix.

The row space of a matrix is the subspace spanned by its row vectors.

The row space of a matrix A is the subspace spanned by its row vectors. In other words, it is the set of all linear combinations of the row vectors of A. The row space is a fundamental concept in linear algebra, as it provides important information about the properties of the matrix.

To find the row space of a matrix, we can use row reduction to put the matrix into row echelon form. The nonzero rows of the resulting matrix form a basis for the row space. Alternatively, we can use the transpose of the matrix and find the column space, which is equivalent to the row space.

The dimension of the row space is equal to the rank of the matrix, which is the number of linearly independent rows (or columns) of the matrix. The row space is also orthogonal to the null space of the matrix, which is the subspace of all solutions to the homogeneous equation Ax=0.

The row space has many applications in linear algebra and beyond. For example, it can be used to determine whether a system of linear equations has a unique solution, or to find the projection of a vector onto a subspace.