The column space of a matrix is the space spanned by the columns of the matrix. It represents all possible linear combinations of the columns. The row space, on the other hand, is the space spanned by the rows of the matrix. It represents all possible linear combinations of the rows.
The basis of a column space/row space provides a minimal set of vectors that span the space. This can be useful in solving systems of linear equations or in finding solutions to other mathematical problems. Additionally, the basis can help in understanding the structure and properties of the matrix.
To find the basis of a column space/row space, we first reduce the matrix to its row-echelon or reduced row-echelon form using elementary row operations. Then, we select the pivot columns/rows as the basis vectors. If there are any non-pivot columns/rows, we add them to the basis. The resulting vectors will form the basis for the column space/row space.
Yes, it is possible for a column space and row space to have different bases. This can happen when the matrix is not square, and therefore, the dimensions of the column space and row space are different. In this case, the basis vectors for the column space and row space will also be different.
The rank of a matrix is equal to the number of linearly independent columns/rows, which is also equal to the number of elements in the basis of the column space/row space. Therefore, finding the basis of a column space/row space can help determine the rank of the matrix, which is an important property in solving systems of linear equations and other mathematical problems.