query: Prove that $\dim(U_{\perp}) = \dim(V ) − \dim(U)$.
positive_passages.docid: 146541
positive_passages.text: Why $\dim U+\dim U^\perp=\dim V$?
negative_passages.docid: 92510
negative_passages.text: Proving a vector space is infinite-dimensional
negative_passages.docid: 150720
negative_passages.text: Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
negative_passages.docid: 62065
negative_passages.text: Prove that $v_1, \dots v_n$ is a basis of V.
negative_passages.docid: 92802
negative_passages.text: Proof of Borsuk-Ulam Theorem .
negative_passages.docid: 149664
negative_passages.text: Norm is sup of inner products (proof).
negative_passages.docid: 59872
negative_passages.text: How to prove $\inf(S)=-\sup(-S)$?
negative_passages.docid: 26658
negative_passages.text: A proof for $\nabla |u|^p = p\ \text{sgn}(u)|u|^{p-1}\nabla u$.
negative_passages.docid: 35564
negative_passages.text: Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$
negative_passages.docid: 132487
negative_passages.text: Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
negative_passages.docid: 81879
negative_passages.text: $Av=λv$ basis proof problems
negative_passages.docid: 19424
negative_passages.text: Showing $(M^\perp)^\perp=\overline{M}$
negative_passages.docid: 53364
negative_passages.text: dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
negative_passages.docid: 39409
negative_passages.text: $X_1,...,X_n$ be independent RVs and $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$. Show that $\sigma (X_1,...,X_n) \perp \mathcal F$
negative_passages.docid: 87494
negative_passages.text: $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
negative_passages.docid: 43504
negative_passages.text: Proving the existence of a complement subspace $U' \subset V$.
negative_passages.docid: 65162
negative_passages.text: Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise.