Questions and Answers
What is the least squares solution?
If there exists an x such that ||Ax-b|| ≈ 0, then x is called the least squares solution.
What is the least squares error formula?
||Ax-b||.
What does the best approximation theorem state?
If W is a finite-dimensional subspace of an inner product space V, and if b is a vector in V, then proj_Wb is the best approximation to b from W.
What is the relationship between Ax and proj_Wb?
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What are the normal equations associated with Ax = b?
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The normal system A^TAx = A^Tb is always consistent.
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What conditions are equivalent for the theorem on A^TA and independence?
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What is the unique least squares solution when A has linearly independent column vectors?
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The normal system A^TA = A^Tb is inconsistent.
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Study Notes
Least Squares Concepts
- Least squares solution: A vector ( x ) exists such that ( ||Ax-b|| \approx 0 ), indicating the best fit for the equation.
- Least squares error: Represented as ( ||Ax-b|| ), this quantity is what is minimized in least squares problems.
Theorems and Projections
- Best approximation theorem: For a vector ( b ) in a vector space ( V ), the projection ( \text{proj}_W b ) is the best approximation from a finite-dimensional subspace ( W ), satisfying ( ||b - \text{proj}_W b|| < ||b - w|| ) for all other ( w ) in ( W ).
- Orthogonal projection: The least squares solution relates to the equation ( Ax = \text{proj}_W b ), indicating how ( b ) is approximated within the subspace ( W ).
Normal Equations
- Normal equation: Linked to the equation ( Ax = b ), it is represented as ( A^T A x = A^T b ). These equations are essential for solving ( Ax = b ) in least squares.
- Normal system theorem: The consistent nature of ( A^T A x = A^T b ) guarantees that every solution to ( A^T b ) is a least squares solution.
Independence and Invertibility
- Invertible ( A^T A ): The conditions that determine this property include the linear independence of ( A )’s column vectors. This ensures ( A^T A ) is square and invertible.
- Linear independence implications: When column vectors of ( A ) are linearly independent, the system ( Ax = b ) yields a unique least squares solution, expressed as ( x = (A^T A)^{-1} A^T b ). The orthogonal projection onto the column space ( W ) is ( \text{proj}_W b = Ax = A(A^T A)^{-1} A^T b ).
Summary of Normal System Properties
- Consistency of normal system: The equation ( A^T A = A^T b ) is consistent, confirming that each solution to ( A^T b ) represents a feasible least squares solution.
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Description
Explore key concepts from Linear Algebra Chapter 6, Part 2 with flashcards focusing on least squares solutions, errors, and the best approximation theorem. Test your understanding and enhance your learning on these essential topics.
