Finding x* in the Nullspace of A: What is its Significance?

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SUMMARY

The discussion centers on finding the point x* in the nullspace of an m×n matrix A, where A has a rank of m and m PREREQUISITES

  • Understanding of linear algebra concepts, specifically nullspaces and projections.
  • Familiarity with matrix rank and its implications in linear transformations.
  • Knowledge of orthogonal projections in R^n.
  • Proficiency in using matrix operations, including transposition and inversion.
NEXT STEPS
  • Study the derivation of the projection formula in linear algebra.
  • Explore the properties of nullspaces and their significance in solving linear equations.
  • Learn about the geometric interpretation of orthogonal projections in vector spaces.
  • Investigate applications of nullspaces in systems of linear equations and optimization problems.
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Students and professionals in mathematics, engineering, and computer science who are interested in linear algebra, particularly those working with matrix theory and optimization techniques.

hoffmann
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Does anyone know how to approach this problem?

Let A be an m×n matrix of rank m, where m<n. Pick a point x in R^n, and let x∗ be the point in the nullspace of A closest to x. Write a formula for x∗ in terms of x and A.

What exactly is the significance of the point x* in the nullspace of A?
 
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i think this question has something to do with the orthogonality of subspaces and is analogous to saying the shortest distance between a point and an axis is just a perpendicular line...
 

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