Linear Algebra Least Squares Question

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SUMMARY

The discussion centers on finding a matrix B such that the projection matrix P = B(B^TB)⁻¹B^T accurately projects onto the column space of matrix A, even when the columns of A are not independent. The standard approach fails due to the non-invertibility of A^TA. Participants suggest considering a basis for the column space of A as a potential solution, rather than rearranging or deleting dependent columns, which could compromise the integrity of the projection.

PREREQUISITES
  • Understanding of projection matrices in linear algebra
  • Familiarity with matrix operations, including transposition and inversion
  • Knowledge of column space and linear independence
  • Basic concepts of basis in vector spaces
NEXT STEPS
  • Research the Gram-Schmidt process for orthogonalization of vectors
  • Study the properties of projection matrices in linear algebra
  • Learn about the Singular Value Decomposition (SVD) and its applications
  • Explore the concept of basis and dimension in vector spaces
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Students and professionals in mathematics, particularly those studying linear algebra, as well as data scientists and engineers working with projections in high-dimensional spaces.

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Homework Statement



Suppose the columns of A are not independent. How could you find a matrix B so that P=B(BTB)^-1BT does give the projection onto the column space of A? (The usual formula will fail when AT A is not invertible).

T is transpose.

Homework Equations


The Attempt at a Solution



I think this is a thought question or something? Do you rearrange the columns...or just delete the dependent columns? But wouldn't that mess up the answer? Idk I got the rest on my p-set but this one I just have nooo idea. I feel like it's really obvious and I'm just missing it. And it's not gram-schmidt or something because that's the section after.
 
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how about considering a basis for the column space?
 

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