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Susanne217
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The basis for the row space of a square matrix is the set of linearly independent rows that span the entire row space. These rows are the minimum number of vectors needed to represent all other rows in the matrix.
The basis for the column space of a square matrix is determined by finding the linearly independent columns that span the entire column space. These columns are the minimum number of vectors needed to represent all other columns in the matrix.
The basis for the row space and column space of a square matrix are related through the concept of duality. This means that the basis for the row space is the transpose of the basis for the column space, and vice versa.
The basis for the null space of a square matrix is comprised of all the vectors that are orthogonal (or perpendicular) to the row space and column space. This means that the basis for the null space is the set of solutions to the homogeneous equation Ax = 0, where A is the square matrix.
Yes, it is possible for the basis of any of these spaces to be empty. This would occur when the matrix has all zero rows or columns, meaning that there are no linearly independent vectors that can span the respective space. In this case, the dimension of the space would be 0.