What Determines Pivot Positions in a Matrix?

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SUMMARY

The discussion centers on determining pivot positions in a matrix, specifically the matrix Ã= and its reduced echelon form. The pivots are confirmed to be in columns 1, 2, and 3, as indicated by the marked positions with X. Additionally, the basis for Col(Ã) and Row(Ã) refers to the set of linearly independent vectors that span the respective spaces, which are indeed the identified pivots. Understanding linear independence is crucial for verifying the basis of these vector spaces.

PREREQUISITES
  • Understanding of matrix operations and reduced row echelon form (RREF).
  • Knowledge of pivot positions and pivot columns in linear algebra.
  • Familiarity with the concept of linear independence in vector spaces.
  • Basic skills in manipulating matrices and performing Gaussian elimination.
NEXT STEPS
  • Study the properties of reduced row echelon form (RREF) in linear algebra.
  • Learn how to determine pivot positions and columns in matrices.
  • Explore the concept of linear independence and how to test for it using determinants.
  • Research the basis of vector spaces, including Col(Ã) and Row(Ã) for various matrices.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching matrix theory and vector spaces.

NorwegianStud
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Matrix: Ã=
1 -2 0 0 3
2 -5 -3 -2 6
0 5 15 10 0
2 6 18 8 6

Matrix on reduced echelon form:
1 0 0 -2 3
0 1 0 -1 0
0 0 1 1 0
0 0 0 0 0

X 0 0 -2 3
0 X 0 -1 0
0 0 X 1 0
0 0 0 0 0

Are the pivot positions the ones I've marked with X? And therefor colum 1, 2 and 3 are pivot columns? Or have I completely misunderstood?
Bonus question: Can't work out "Find a basis for Col(Ã), Row(Ã)". What does that even mean?
 
Last edited:
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Those are indeed the pivots.

As for the bonus, the basis is just a set of linearly independent vectors that can represent every vector in a space.
 
Last edited:
QuarkCharmer said:
Those are indeed the pivots.

As for the bonus, the basis is just a set of linearly independent vectors that can represent every vector in a space.

Great, thanks a lot.

Linearly independent vectors? I've seen that before when I looked for the answer, but I didn't quite understand it. How do I check if it's linearly independent?
 

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