Difference among types of bases.

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    Bases Difference
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Discussion Overview

The discussion revolves around the concept of bases in linear algebra, specifically addressing the confusion regarding the types of bases associated with a matrix A. Participants explore the implications of finding a basis for A, distinguishing between bases for the null space, range, column space, and row space.

Discussion Character

  • Conceptual clarification, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses confusion about what is meant by finding a basis for A, questioning whether it refers to any basis related to the null space or range.
  • Another participant suggests visualizing the set of matrices as a vector space and provides an example of a basis for a specific subspace of matrices.
  • A different participant argues that if A is a matrix, the question lacks meaning, while if A is a vector space, it could be valid.
  • Another participant proposes that the question likely refers to finding a basis for the column-space or row-space of A.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of the question regarding finding a basis for A, with multiple competing views presented about what this could entail.

Contextual Notes

Participants highlight the ambiguity in the question, particularly regarding the definitions of A and the context in which the basis is being sought.

smithnya
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Hello everyone,

I am currently learning about bases, and I am confused about different types of bases. I understand how to obtain a basis for the null space of A or a basis for the range of A. I recently ran into a problem that simply asked me to find a basis for A. What does this mean? Does it mean any basis (for N(A) or R(A))? OR is there such a thing as a separate basis for A not having to do with R(A) or N(A)?
 
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Maybe it wants you to visualize the set of matrices as a vector space?

Consider for example the set of matrices of the form

|a 0|
|0 b|

They are spanned by the two independent elements

|1 0|
|0 0|

|0 0|
|0 1|

so this is a basis for the subspace of matrices given above. What is the exact form of A and of the question?
 
If A is a matrix, the question is nonsense.

If A is a vector space, then it makes sense.
 
my guess is what is meant is a basis for the column-space or row-space of A.
 

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