How to tell a dimension of a subspace

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How to tell the dimension of a subspace?

I know it is depending on the numbers of linear independent basis. But in the question, it already has two independent basis, and I can already think of a lot more independent basis other than $u$ and $v$..

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The dimension theorem for vector spaces states that all bases of a vector space have the same number of elements. So if you can determine the cardinality of one basis, you have determined the dimension.