Using Dimension Formula of a Matrix

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Homework Help Overview

The discussion revolves around a problem in linear algebra concerning the dimensions of vector spaces, specifically focusing on the relationship between the dimensions of the sum and intersection of two vector spaces, V and W, in the context of a matrix A of rank r.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply the dimension formula for vector spaces but expresses uncertainty about how to begin. They mention a related equation involving the rank and nullity of a matrix.
  • Some participants question the notation used, particularly the meaning of the symbol \Omega and clarify that V and W are vector spaces rather than just vectors.

Discussion Status

The discussion is ongoing, with participants seeking clarification on terminology and the original poster looking for guidance on how to approach the problem. There is no explicit consensus yet, but the clarification of terms is a productive step forward.

Contextual Notes

There is a lack of clarity regarding the definitions of the symbols used in the problem statement, which may affect the understanding of the dimension formula being discussed.

tatianaiistb
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Homework Statement


If VΩW contains only the zero vector, then dim(V+W)+dim(VΩW) = dim(v) + dim(W) becomes dim(V+W) = dim(v) + dim(W). Check this when V is the row space of A, W is the nullspace of A, and the matrix A is mxn of rank r. What are the dimensions?


Homework Equations



dim(V+W)+dim(VΩW) = dim(v) + dim(W)

The Attempt at a Solution



I don't even know how to start this problem.

I do know that dim(V+W) + dim (VΩW) = rank of [A B] + nullity of [A B]

Can anyone help? Thanks!

 
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It would help if you told us what thes symbols mean. I presume that V and W are vectors but what is [itex]\Omega[/itex] and what do you mean by [itex]V\Omega W[/itex]?
 
I'm sorry! That symbol means V intersects W... And yes, V and W are vectors.
 
tatianaiistb said:
I'm sorry! That symbol means V intersects W... And yes, V and W are vectors.
No, V and W are vector spaces.
 

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