Confused about col/rows of matrices

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SUMMARY

This discussion clarifies the placement of vectors in matrices when performing Row Reduced Echelon Form (RREF) operations. Specifically, it establishes that in an m x n matrix, vectors representing a subspace are placed in columns when they correspond to the target space k^m and in rows when they represent linear components of the map from k^n to k. The distinction is crucial for understanding linear mappings and their representation in matrix form.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly linear mappings.
  • Familiarity with Row Reduced Echelon Form (RREF) techniques.
  • Knowledge of vector spaces and their dimensions.
  • Basic proficiency in matrix representation and operations.
NEXT STEPS
  • Study the properties of linear maps in linear algebra.
  • Learn about the implications of matrix dimensions in transformations.
  • Explore examples of RREF with different vector placements.
  • Investigate the relationship between row and column spaces in matrices.
USEFUL FOR

Students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of vector placement in matrix operations.

PragmaOnce
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Can someone tell me (when RREFing a matrix) when do we put the vectors of a subspace in columns of a matrix and when in rows?

Example from my notes:
Here, my prof put the vectors in columns:
25ge2gy.png


and here, he put the vectors in rows:
8wgm8o.png


Thanks!
 
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an mxn matrix represents a linear map from k^n to k^m. the rows represent linear components functions of the map, i.e. linear maps from k^n to k, and there are m of them. the columns represent vectors in the target space k^m, and there are n of them. they are the values of the map on the n standard basis vectors of the source space.
 

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