jazz_hole said:
jazz_hole said:
A matrix that is n × n has the same number of rows and columns, with n representing the size of the matrix. This means that both A and B have the same number of rows and columns, with each matrix having n rows and n columns.
The range of a matrix is the set of all possible output values that can be obtained by multiplying the matrix with a vector. In other words, it is the set of all possible solutions to the equation Ax = b, where A is the matrix and b is a vector.
It is important to show that range(AB) ⊆ range(A) because it helps to understand the relationship between the two matrices and how their ranges are related. It also helps to determine if the range of AB is a subset of the range of A, which can have important implications in solving systems of linear equations.
To prove that range(AB) ⊆ range(A), we can use the fact that range(AB) is a subset of the column space of A, and that the column space of A is equal to the range of A. Therefore, any vector in the range of AB can also be found in the range of A, showing that range(AB) ⊆ range(A).
Yes, it is possible for range(AB) to be equal to range(A). This can happen when the columns of B are linear combinations of the columns of A, or when the columns of A and B are linearly dependent. In these cases, the range of AB will be the same as the range of A.