Assertion: matrix's rank | /r/HasProperty | equal
Facets:
value: to r
type: OTHER
value: to its rank
type: OTHER
value: to the rank of its transpose
type: OTHER
value: to n
type: OTHER
Source sentences:
text: If A is m-by-n and the rank of A is equal to n (
text: The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix.
text: It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the determinantal rank of a matrix is equal to its rank).
text: The rank of a matrix is also equal to the dimension of the column space.
text: Indeed, since the column vectors of A {\displaystyle A} are the row vectors of the transpose of A {\displaystyle A} , the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e., rank ⁡
text: Now apply this result to A ∗ {\displaystyle A^{*}} to get the reverse inequality and conclude that the ranks are equal.
text: The rank of A is equal to r if and only if there exists an invertible m × m matrix X and an invertible n × n matrix Y such that
text: If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are equal.
text: If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are equal.
text: If, on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution.