peripatein said:Why are the m vectors in R^n?
The row space of a matrix is the span of the rows of the matrix. It is a subspace of the vector space that the matrix operates on, and it represents all possible linear combinations of the rows of the matrix.
The row space and column space of a matrix are related through the rank of the matrix. The rank of a matrix is equal to the dimension of both the row space and the column space. This means that the row space and column space are essentially two different perspectives on the same subspace of the vector space.
The row space is important in matrix operations because it helps us understand the fundamental properties of matrices, such as their rank, determinant, and inverse. It also allows us to determine the solutions to systems of linear equations through methods like row reduction.
The row space can be calculated by finding the linearly independent rows of a matrix, which form a basis for the row space. This can be done through methods like Gaussian elimination or by finding the reduced row echelon form of the matrix.
No, the row space and null space of a matrix cannot be equal. The row space contains all possible linear combinations of the rows of the matrix, while the null space contains all vectors that are mapped to the zero vector by the matrix. These two subspaces are only equal in the case of the zero matrix, where both the row space and null space are trivial.