What Does It Mean When Col A Is a Subspace of the Null Space of A?

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SUMMARY

The discussion clarifies that when the column space (Col A) of a matrix A is a subspace of the null space (Null Space of A), it indicates that for any vector y in Col A, the equation A(Ax) = 0 holds true, leading to the conclusion that A² = 0. This relationship necessitates that A is a square matrix, as it maps a vector space U to itself. Additionally, the null space of A transpose multiplied by A (A^T A) is also explored, emphasizing the importance of understanding these concepts in linear algebra.

PREREQUISITES
  • Understanding of linear transformations and vector spaces
  • Familiarity with matrix operations and properties
  • Knowledge of column space and null space definitions
  • Basic concepts of square matrices in linear algebra
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  • Research the properties of square matrices and their implications in linear algebra
  • Study the relationship between column space and null space in detail
  • Learn about the implications of A² = 0 in matrix theory
  • Explore theorems related to null spaces, such as the Rank-Nullity Theorem
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quantumlight
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I am just wondering what is meant when someone says the Col A is a subspace of null Space of A. What can you infer from that?

Also what is a null space of A(transpose)A

How do they relate to A? Are there theorems about this that I can look up?
 
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First, if A maps vector space U to vector space V, the column space of A is a subset of and the null space of A is a subset of U so in order for that to makes sense U and V must be the same: A maps a space U to itself. In terms of matrices, that means A must be a square matrix. The columns space is the "range" of A. If y is in the column space of A, that means there exist some x such that Ax= y. If y is also in the null space, then Ay= A(Ax)= 0. Finally, if the column space is a subset of the null space, that must always be true: A(Ax)= A2x= 0. Again, in terms of matrices that means A2= 0.
 

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