query: Is the formula $(\text{ker }A)^\perp=\text{im }A^T$ necessarily true?
positive_passages.docid: 45998
positive_passages.text: The range of $T^*$ is the orthogonal complement of $\ker(T)$
negative_passages.docid: 11557
negative_passages.text: How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
negative_passages.docid: 61538
negative_passages.text: Given a square matrix $A$, both $AA^T$ and $A^TA$ are symmetric
negative_passages.docid: 14161
negative_passages.text: Proof of when is $A=X^TX$ invertible?
negative_passages.docid: 155884
negative_passages.text: What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true?
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
negative_passages.docid: 53899
negative_passages.text: Proof of the Sherman-Morrison Formula
negative_passages.docid: 24072
negative_passages.text: Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$
negative_passages.docid: 67784
negative_passages.text: If $A + A^t = 2I$ then $\det(A) \geq 1$
negative_passages.docid: 359
negative_passages.text: Show that $\ker \hat{T} = \text{ann}(\text{range } T)$
negative_passages.docid: 155828
negative_passages.text: How do I show that $A$ and $A^T$ are similar, given that the minimal polynomial of $A$ splits over $\mathbb{F}$?
negative_passages.docid: 67058
negative_passages.text: Prove that $\text{rank}(X^TX)=\text{rank}(X)$
negative_passages.docid: 83475
negative_passages.text: Prove Or Disprove: tr(AB)=tr(A)*tr(B)
negative_passages.docid: 147207
negative_passages.text: Is it always true that $\det(A^2+B^2)\geq0$?
negative_passages.docid: 6169
negative_passages.text: If $ A^3=A$ prove that $Ker\left(A-I\right)+Im\left(A-I\right)=V$
negative_passages.docid: 129647
negative_passages.text: $A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$
negative_passages.docid: 54294
negative_passages.text: Can the product $AB$ be computed using only $+, -,$ and reciprocal operators?