query: prove that every generating set of a vector space $V$ has a minimal generating set
positive_passages.docid: 151467
positive_passages.text: An infinite generating set of a finite dimensional vector space contains a basis
negative_passages.docid: 4151
negative_passages.text: Every subspace of a vector space has a complement
negative_passages.docid: 46265
negative_passages.text: Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$
negative_passages.docid: 119782
negative_passages.text: Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.
negative_passages.docid: 102371
negative_passages.text: Proving that two vector spaces of equal dimension are isomorphic
negative_passages.docid: 5347
negative_passages.text: Using the definition of a vector space to prove that $0x = 0$ and $(-1)x = -x$
negative_passages.docid: 43503
negative_passages.text: Bases of complex vector spaces and the axiom of choice
negative_passages.docid: 52596
negative_passages.text: A basis for $k(X)$ regarded as a vector space over $k$
negative_passages.docid: 125785
negative_passages.text: Minimal polynomial of restriction to invariant subspace divides minimal polynomial
negative_passages.docid: 54737
negative_passages.text: Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$
negative_passages.docid: 66500
negative_passages.text: Proving that orthogonal vectors are linearly independent
negative_passages.docid: 129810
negative_passages.text: What kind of vector spaces have exactly one basis?
negative_passages.docid: 47015
negative_passages.text: Vector Space Structures over ($\mathbb{R}$,+)
negative_passages.docid: 21820
negative_passages.text: How to prove that a set is invariant?
negative_passages.docid: 34099
negative_passages.text: Proving that the empty set is unique
negative_passages.docid: 132492
negative_passages.text: If a linear transformation $T$ has $z^n$ as the minimal polynomial, there is a vector $v$ such that $v, Tv,\dots, T^{n-1}v$ are linearly independent
negative_passages.docid: 137873
negative_passages.text: Problems that involves two facts in the definition of vector spaces