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| 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. | 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]] (''R'') looks like the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|identity matrix]] (''I'') with some number of zero rows. |
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| If a square matrix has '''full rank''', the [[LinearAlgebra/Elimination#Reduced_Row_Echelon_Form|reduced row echelon form]] of the matrix is the [[LinearAlgebra/SpecialMatrices#Identity_Matrix|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 square matrix has '''full rank''', the reduced row echelon form (''R'') of the matrix is the identity matrix (''I''). 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''. |
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 (R) looks like the identity matrix (I) 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 (R) of the matrix is the identity matrix (I). 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.