Proving Linear Algebra Concepts: Rank, RREF, Invertibility, and Dependency

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SUMMARY

This discussion focuses on proving key concepts in linear algebra, specifically regarding matrix rank, reduced row echelon form (RREF), invertibility, and linear dependence. Participants are tasked with finding matrices A and B where Rank [AB]≠Rank(BA), identifying a matrix A such that Rref(A)≠Rref(A^T), solving for X in the equation BXB^-1 – A = I_n, and demonstrating the linear dependence of the matrix formed by vectors Ab_1, Ab_2, and Ab_3 given that {b1, b2, b3} is linearly dependent. The consensus is that problems 1 and 2 require experimentation with different matrices, while problem 3 involves algebraic manipulation, and problem 4 necessitates applying the definition of linear dependence.

PREREQUISITES
  • Understanding of matrix rank and properties of matrix multiplication.
  • Familiarity with reduced row echelon form (RREF) and matrix transposition.
  • Knowledge of invertible matrices and the identity matrix.
  • Concept of linear dependence and its mathematical definition.
NEXT STEPS
  • Explore matrix multiplication properties and their implications on rank.
  • Study the process of obtaining reduced row echelon form (RREF) for various matrices.
  • Learn about the conditions for matrix invertibility and how to solve matrix equations involving inverses.
  • Investigate the definition of linear dependence and practice proving dependence with different sets of vectors.
USEFUL FOR

Students and educators in linear algebra, mathematicians, and anyone looking to deepen their understanding of matrix theory and linear dependence concepts.

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1) Find two matrices A and B where Rank [AB]≠Rank(BA)

2) Find a matrix A where Rref(A)≠Rref(A^T) where T is the transpose

3) Find X given that B is invertible if BXB^-1 –A=I_n (identity matrix)

4) Prove that [Ab_1 Ab_2 Ab_3] is linearly dependent given that {b1 b2 b3} is linearly dependent.

i can't get any of these and tried substituting numbers and nonzero rows and columns to obtain any of the four. Can someone please help me get these? Thank you to those who help in advance!
 
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1-2 is just a matter of trying more matrices. 3 is just algebra. For 4 you should use the definition of linear dependence.
 

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