Assertion: matrix's column | /r/HasProperty | independent
Facets:
value: linearly
type: DEGREE
value: in the full matrix
type: OTHER
value: by reducing only to echelon form
type: MANNER
Source sentences:
text: The columns of A are linearly independent.
text: From the definition of the parity-check matrix it directly follows the minimum distance of the code is the minimum number d such that every d - 1 columns of a parity-check matrix H are linearly independent while there exist d columns of H that are linearly dependent.
text: At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two.
text: This makes it possible to determine which columns are linearly independent by reducing only to echelon form.
text: has rank 2: the first two columns are linearly independent, so the rank is at least 2, but since the third is a linear combination of the first two (the second subtracted from the first), the three columns are linearly dependent so the rank must be less than 3.
text: A non-vanishing p-minor (p × p submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the converse is less straightforward.
text: A non-vanishing p-minor (p × p submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the converse is less straightforward.