MAT 211 : Linear Algebra

If A is not invertible, then Ax=b has one of two problems. For almost all values of b, the system is overconstrained, and x has no exact solution. For few values of b (including b=0) the system is underconstrained, and x has infinitely many solutions.

Rank-Nullity Theorem
#PivotColumns
#IndependentColumns(A)
Dim(ColumnSpace(A))
Dim(Im(A))
Rank A
+ #FreeColumns
#DependentColumns(A)
Dim(Null(A))
Dim(Ker(A))
Nullity A
= #Columns(A)
n