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Here's what I understand from Basis.

Basis: it's set of linearly independent vectors which can span the vector space.

Basis for Zero vector space:

case 1: when { 0 } , it's singleton set , since there's none in it to compare with to check linear independecy hence it can be considered as basis and also it can span it.

( i.e 0 = c . 0 , where c is any constant )

So, the basis has 1 vector, so dimension will be 1.

case 2: { any vector K } again it's singleton set , since there's none in it to compare with to check linear independecy hence it can be considered as basis and also it can span it.

( i.e 0 = 0. K (the vector K) )

Again, the basis has 1 vector, so dimension will be 1.

Inception
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    Short answer: Because its basis is the empty set $\emptyset$. See the duplicate for more details. – Dietrich Burde Feb 04 '24 at 10:06
  • Personally, I consider the zero vector space having dimension zero to be a convention. To actually make it follow from definition, you need to be more careful in how you define basis. – David Gao Feb 04 '24 at 10:07
  • @DietrichBurde I appreciate your response. can you also please answer where I am wrong in my argument. Thanks – Inception Feb 04 '24 at 10:07
  • You are wrong because of what egreg says in his answer. You forgot case $3$, the only one which works (namely What's the largest linearly independent set in ${\mathbf{0}}$? The only subsets in it are the empty set and the whole set.) – Dietrich Burde Feb 04 '24 at 10:08
  • Bases are linearly independent. So, in order for ${\vec{0}}$ to be a basis, we must have $c \vec{0} = \vec{0} \implies c = 0$. But this is not true, for example if $c = 1$. – Theo Bendit Feb 04 '24 at 11:08

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