query: Prove that two bases for a vector space are equinumerous
positive_passages.docid: 27478
positive_passages.text: Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
negative_passages.docid: 87692
negative_passages.text: Basis and dimension of vector space
negative_passages.docid: 133611
negative_passages.text: Intersection of two vector spaces
negative_passages.docid: 21807
negative_passages.text: on proving bac-cab rule, $\vec{A} \times ( \vec{B} \times \vec{C})= \vec{B} (\vec{A} \cdot\vec{C})- \vec{C}(\vec{A}\cdot\vec{B})$
negative_passages.docid: 120535
negative_passages.text: Basis of the polynomial vector space
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: 57649
negative_passages.text: Prove that for every $f$, a bilinear form, there exists a basis ${v_1,...,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$
negative_passages.docid: 129810
negative_passages.text: What kind of vector spaces have exactly one basis?
negative_passages.docid: 81879
negative_passages.text: $Av=λv$ basis proof problems
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: 95206
negative_passages.text: If $\{u,v,w\}$ is a basis for $V$, then $\{u+v+w,v+w,w\}$ is also a basis: is this proof correct?
negative_passages.docid: 723
negative_passages.text: Is there a constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$?
negative_passages.docid: 152742
negative_passages.text: Finding a basis of an infinite-dimensional vector space?
negative_passages.docid: 68526
negative_passages.text: Polynomials as vector spaces?
negative_passages.docid: 158156
negative_passages.text: Finding the dual basis for the space of polynomials of degree $\leq$ 2
negative_passages.docid: 13880
negative_passages.text: Example of vector space such that $cv = 0$ where $c \ne 0$ and $v \ne \vec{0}$
negative_passages.docid: 70475
negative_passages.text: Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.