a1234 said:I'm doing problems on finding row and column spaces. My textbook tells me to find the echelon form of the matrix, and then to identify the bases. My question is, can I reduce the matrix to reduced echelon form to get the bases? I have the same question about bases for the solution space.
The purpose of finding bases for row and column spaces is to identify a set of vectors that can be used to span the entire space of a matrix. These bases are important in linear algebra as they help in solving systems of linear equations, determining the rank of a matrix, and understanding the properties of a matrix.
To find bases for row and column spaces, you can use the row reduction method or the Gaussian elimination method. Both methods involve performing elementary row operations on a matrix to reduce it to its row echelon form. The nonzero rows of the row echelon form will form the basis for the row space, while the pivot columns will form the basis for the column space.
Yes, a matrix can have multiple bases for its row and column spaces. This is because there can be different sets of linearly independent vectors that can span the same space. However, all bases for a particular space will have the same number of vectors, known as the dimension of the space.
The row space and the column space of a matrix are related by the rank of the matrix. The rank of a matrix is the number of linearly independent rows or columns in the matrix. The dimension of the row space and the column space will always be equal to the rank of the matrix.
No, a matrix cannot have a different number of bases for its row and column spaces. This is because the dimension of the row space and the column space will always be equal to the rank of the matrix. Therefore, the number of bases for both spaces will also be the same.