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for system | for system | ||
| − | :<math>A\bold{x}=\bold{b}</math> | + | :<math>A\bold{x}=\bold{b},</math> |
| − | :<math> {A^T(A\bold\hat{x} - \bold{b})} = 0 </math> | + | :<math> {A^T(A\bold{\hat{x}} - \bold{b})} = 0,</math> |
which is equivalent to | which is equivalent to | ||
| − | :<math> {A^TA\bold\hat{x}} = A^{T}\bold{b} </math> | + | :<math> {A^TA\bold{\hat{x}}} = A^{T}\bold{b}.</math> |
| Line 15: | Line 15: | ||
Thorough examples are available in the [[MA265]] textbook, Webassign problems, and Past Exam Archive. Most problems only deal with linear and parabolic fits. | Thorough examples are available in the [[MA265]] textbook, Webassign problems, and Past Exam Archive. Most problems only deal with linear and parabolic fits. | ||
| + | |||
| + | |||
| + | '''Main Reference''' | ||
| + | ---- | ||
| + | Kolman, B., & Hill, D. (2007). ''Elementary linear algebra with applications (9th ed.)''. Prentice Hall. | ||
Latest revision as of 21:37, 4 March 2015
The Least Squares Solution
The Least Squares Approximation is a examples-intensive concept. However, it can be solved using the following concise formulas:
for system
- $ A\bold{x}=\bold{b}, $
- $ {A^T(A\bold{\hat{x}} - \bold{b})} = 0, $
which is equivalent to
- $ {A^TA\bold{\hat{x}}} = A^{T}\bold{b}. $
NOTE:
Thorough examples are available in the MA265 textbook, Webassign problems, and Past Exam Archive. Most problems only deal with linear and parabolic fits.
Main Reference
Kolman, B., & Hill, D. (2007). Elementary linear algebra with applications (9th ed.). Prentice Hall.
Ryan Jason Tedjasukmana
Back to Inner Product Spaces and Orthogonal Complements
