query: Why does a basis of a vector space need to have the same number of vectors as the vector space dimension?
positive_passages.docid: 81379
positive_passages.text: Any two bases of a finite dimensional vector space must have the same number of elements.
negative_passages.docid: 154724
negative_passages.text: Every vector space has a basis
negative_passages.docid: 124042
negative_passages.text: Why is a vector space equal to its tangent space for any point?
negative_passages.docid: 68526
negative_passages.text: Polynomials as vector spaces?
negative_passages.docid: 122623
negative_passages.text: Geometric intuition for the tensor product of vector spaces
negative_passages.docid: 152742
negative_passages.text: Finding a basis of an infinite-dimensional vector space?
negative_passages.docid: 92510
negative_passages.text: Proving a vector space is infinite-dimensional
negative_passages.docid: 15020
negative_passages.text: How to count number of bases and subspaces of a given dimension in a vector space over a finite field?
negative_passages.docid: 18487
negative_passages.text: Why are there only $n-1$ principal components for $n$ data if the number of dimensions is $\ge n$?
negative_passages.docid: 156985
negative_passages.text: Question Concerning Vectors
negative_passages.docid: 76160
negative_passages.text: Dimension of the sum of two vector subspaces
negative_passages.docid: 151467
negative_passages.text: An infinite generating set of a finite dimensional vector space contains a basis
negative_passages.docid: 38516
negative_passages.text: Show a subset is subspace of a vectorspace and find the dimension
negative_passages.docid: 14429
negative_passages.text: dimension of a subspace spanned by two subspaces
negative_passages.docid: 43814
negative_passages.text: What does it mean when a set is said to be a "finitely generated vector space"?
negative_passages.docid: 110654
negative_passages.text: Vector Spaces and AC
negative_passages.docid: 78833
negative_passages.text: Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?