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Let $$V = \{ v \in M_{33}| \text{ equal column sums} \}$$ be a vector space.

How would you calculate

$$P( \text{ equal column sums} | v \in M_{33} )$$

?

How would you go about solving these types of problems in general?

RaoulConstantine
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  • Can you say what P() means? I guess it is a probability but of what and what is the sample space? – Paul Jan 25 '24 at 16:25
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    $V$ is the set of matrices with equal column sums and you are wondering about the probability that a matrix has equal column sums, given you know it is a matrix from the set $V$? This is just $100%$? – Student Jan 25 '24 at 16:32
  • No, V is the set of matrices with equal column sums but the question is about V as subset or [math]M_{33}[/math]. – George Ivey Jan 25 '24 at 17:18
  • You haven't specified a distribution over $M_{33}$, but for any reasonable distribution, the probability that the column sums are all equal is $0$. – joriki Jan 25 '24 at 17:20
  • Sample space is all 3 x 3 matrices, I want to know the probability of choosing a matrix such that the column sums are equal. So this probability is equal to zero? Because the set of 3 x 3 matrices has cardinality much greater than the set of 3 x 3 matrices with equal column sums? I suppose the distribution I had in mind is that each element has equal probability. – RaoulConstantine Jan 25 '24 at 17:26
  • @RaoulConstantine: As to the distribution, please see Why isn't there a uniform probability distribution over the positive real numbers?. As to your question about cardinality: No, it's not a question of cardinality but of measure. The condition that the column sums are equal reduces the dimension, and sets of lower dimension have measure zero in the higher-dimensional space. – joriki Jan 25 '24 at 17:32
  • Ok that’s really interesting and has lead to good further reading on measure theory. For example this thread helped https://math.stackexchange.com/questions/618340/sub-dimensional-linear-subspaces-of-mathbbrn-have-measure-zero

    Can you recommend any other further reading or threads? Thanks

     – RaoulConstantine Jan 25 '24 at 18:39

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