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What is the maximum dimensionality of the linear manifold spanned by:

a) 20 points in a 40 dimensional space b) 40 points in a 20 dimensional space c) 20 points in a 20 dimensional space 

It has been a long time since I took the linear algebra course, so please correct me if I am wrong.

My answers:

a) 40 b) 20 c) 20 
  • 1
    A simple way to see why your first answer is wrong: to have a 40-dimensional vector space, you need (at least) 40 vectors to form a basis. With 20 points, at best you can have a basis of cardinality 20.2011-05-13

1 Answers 1

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The maximum dimension $d$ of the linear manifold $M$ spanned by $m$ points in a $n$-dimensional vector space $V$ is $d=\min(m,n)$.

Indeed, $M$ is a submanifold of $V$, so $d\le n$, and $M$ is spanned by $n$ points, so $d\le m$, hence $d\le \min(m,n)$. Conversely, since $n=\dim V$ there are $n$ linearly independent vectors in $V$; take $\min(m,n)$ of them (and throw in $m-n$ copies of the zero vector if m>n) and then take the span to get a $\min(m,n)$-dimensional manifold $M$ spanned by $m$ points, showing that $d\ge \min(m,n)$.

So the answer to each of your questions should be $20$.