query: Prove that $ \text {(null}~T^*)^\perp \subseteq \text{range}~T $
positive_passages.docid: 45998
positive_passages.text: The range of $T^*$ is the orthogonal complement of $\ker(T)$
negative_passages.docid: 52614
negative_passages.text: (exercise from Tao's analysis book) Proof of a lemma relating to power set of X
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$
negative_passages.docid: 55488
negative_passages.text: Proving that $\operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta})$
negative_passages.docid: 72654
negative_passages.text: If $(a_n)$ is a null sequence and $(b_n)$ is bounded, then $(a_nb_n)$ is a null sequence
negative_passages.docid: 142229
negative_passages.text: Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 94077
negative_passages.text: Proof of linear independence of $e^{at}$
negative_passages.docid: 27368
negative_passages.text: (Verification) $C \approxeq A$ or $C \approxeq \text{section of } A$ $\forall C$ s.t. $C \subset A$
negative_passages.docid: 23580
negative_passages.text: Range of A and null space of the transpose of A
negative_passages.docid: 145447
negative_passages.text: bilinear form - proof
negative_passages.docid: 115533
negative_passages.text: Inequality for small values of $t$
negative_passages.docid: 95266
negative_passages.text: Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$
negative_passages.docid: 10483
negative_passages.text: Prove Sylvester rank inequality: $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n$
negative_passages.docid: 16789
negative_passages.text: Prove that $T^2$ is the zero transformation if and only if $\mathrm{range}(T) \subseteq \ker(T)$.
negative_passages.docid: 108704
negative_passages.text: Help with proof of $|x+y| \geq |x|-|y|$
negative_passages.docid: 91022
negative_passages.text: How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?