Find the least square solutions of the system where
The solution is .
Consider a linear system as .
Here A is an matrix, a vector in called a least square solution of this system if role="math" localid="1659687531916" .
Consider a linear system as where and A also A is A is an matrix, a vector in called a least square solution of this system if
The term least square solution reflects the fact that minimizing the sum of the squares of the component of the vector .
To show of a linear system .
Consider the following string of equivalent statements.
The vector is a least square solution of the system .
Then by the definition as follows.
Also by the theorem 5.4.3 as follows.
Similarly by the theorem 5.1.4 and 5.4.1 as follows.
Therefore, the least square of the system are the exact solution of the system .
The system is called the normal equation of .
Now, simplify as follows.
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