Can You Find a Matrix Where Column Space Equals Null Space?

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SUMMARY

The discussion centers on finding a 4 x 4 matrix A where the column space (col A) equals the null space (null A). It is established that for this condition to hold, both dimensions must equal 2, as defined by the equation dim(col A) + dim(null A) = 4. The key insight is that if col A = null A, then A squared (A²) must equal the zero matrix, indicating that A is nilpotent. Therefore, a nilpotent matrix of size 4 x 4 with a rank of 2 serves as an example.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically column space and null space.
  • Familiarity with matrix dimensions and rank.
  • Knowledge of nilpotent matrices and their properties.
  • Basic proficiency in matrix operations, including multiplication.
NEXT STEPS
  • Research the properties of nilpotent matrices in linear algebra.
  • Learn how to compute the rank and nullity of matrices.
  • Explore examples of 4 x 4 nilpotent matrices.
  • Study the implications of the Rank-Nullity Theorem in depth.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking for examples of nilpotent matrices and their applications.

iNagib
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Hey fellas could use a hand on a linear question that has been bugging me.

Give an example of a 4 x 4 matrix such that col A = nul A
Appreciate any help!
Thanks,
ib
 
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To start with, remember that dim(col A) + dim(null A) = 4, so we better have dim(col A) = dim(null A) = 2.
 
If col A = null A, then given any vector x in R4, Ax ∈ col A = null A, so A2x = 0. Since x is arbitrary, A2 = 0. Conversely, if A2 = 0, you can show that col A ⊆ null A, and you need dim(col A) = 2. This might help a bit in finding such a matrix.

(Matrices A such that Ak = 0 for some integer k are called nilpotent.)
 

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