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- From: kbanerje@grad09.math.ncsu.edu (Kaushik Banerjee)
- Subject: Re: simultaneous equations...solutions
- Message-ID: <1993Jan11.201751.18414@ncsu.edu>
- Sender: news@ncsu.edu (USENET News System)
- Organization: NCSU Mathematics Dept.
- References: <1993Jan10.191952.16384@nuscc.nus.sg> <1993Jan10.211732.24254@ncsu.edu>
- Date: Mon, 11 Jan 1993 20:17:51 GMT
- Lines: 33
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- CORRECTION to my previous post:
-
- Fact #1
-
- An nxn linear system Ax=b has a unique solution if
- and only if A is nonsingular.
-
- Fact #2
-
- An mxn (where m .ne. n) linear system Ax=b has either
- no solution (termed "inconsistent") or an infinite number
- of solutions.
-
- Fact #3
-
- The best you can do is a "least squares" solution.
- Notation: A' = transpose of A
- A^ = inverse of A
- ||x|| = Euclidean norm of the vector x
- Say you want to solve Ax=b where A is nx3
- ==> A'Ax = A'b (these are the "normal equations")
- If the columns of A are linearly independent, then the
- 3x3 matrix A'A is nonsingular.
- ==> x = (A'A)^A'b, i.e., x is the unique solution to the
- normal equations and the unique least squares solution
- to Ax=b.
-
- Definition of a least squares solution:
- x is least squares solution if and only if
- ||Ax - b|| .le. ||Az - b|| for all other vectors z.
- ^ ---> NOTE this correction.
-