When a matrix
A is square with full rank, there is a vector
that satisfies the equation
for any
. However, when
A is not square or does not have full rank, such an
may not exist, because
b does not lie in the range of
A. In this case, called the least squares problem, we seek the vector
x that minimizes the length (or
norm) of the residual vector
. The four vectors
,
,
, and
are color coded and the plane is the range of the matrix
. The plane shown is the set of all possible vectors
.
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