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LongApple
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I see that the pivots in columns 1 and 2 help us decide which columns to take. But why does the L matrix of this B = LU let just to read off the column space?
2:18
2:18
The L matrix from LU decomposition represents the lower triangular matrix which is responsible for the elimination of variables in the system of linear equations. This elimination process results in a reduced row echelon form matrix, which in turn, gives us the column space of the original matrix.
L represents the lower triangular matrix in LU decomposition and plays a crucial role in the elimination process. It helps in reducing the original matrix to its reduced row echelon form, which gives us the column space of the original matrix.
Yes, we can use LU decomposition to find the column space of any matrix. This method is particularly useful in solving systems of linear equations, as it helps in reducing the matrix to its reduced row echelon form, which gives us the column space.
The L matrix from LU decomposition helps in finding linearly independent columns by eliminating the dependent columns in the original matrix. The reduced row echelon form matrix obtained after the elimination process has the same number of linearly independent columns as the original matrix, making it easier to identify them.
One limitation of using LU decomposition to find the column space is that the original matrix must be invertible. If the matrix is not invertible, the elimination process will not be possible, and the column space cannot be determined. Additionally, LU decomposition can be computationally expensive for larger matrices, making it less practical for certain applications.