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| If a matrix has a pivot in each column, it is said to be '''full column rank''' and the only [[LinearAlgebra/NullSpaces|null space]] is the zero vector (i.e. ''[0 ... 0]''). ''Ax = b'' either has one solution or is not solvable. Incidentally, the [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reduced row echelon form]] looks like the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] with some number of zero rows. | If a matrix has a pivot in each column, it is said to be '''full column rank'''. The only [[LinearAlgebra/NullSpaces|null space]] is the zero vector (i.e. ''[0 ...]''). ''Ax = b'' either has one solution or is not solvable. Incidentally, the [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reduced row echelon form]] looks like the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] with some number of zero rows. |
Rank
The rank of a matrix is the number of pivots.
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If a matrix has a pivot in each column, it is said to be full column rank. The only null space is the zero vector (i.e. [0 ...]). Ax = b either has one solution or is not solvable. Incidentally, the reduced row echelon form looks like the identity matrix with some number of zero rows.
If a matrix has a pivot in each row, it is said to be full row rank. This only means that Ax = b can be solved for any b.
If a square matrix has full rank, the reduced row echelon form of the matrix is the identity matrix. The only null space is the zero vector; Ax = b can be solved for any b; there is exactly one solution for any given b.
If a matrix is none of the above, Ax = b either can be solved for any b or is not solvable.