query: I need to show that $\operatorname{ker}(A^T) = (\operatorname{Im} A)^\perp$
positive_passages.docid: 45998
positive_passages.text: The range of $T^*$ is the orthogonal complement of $\ker(T)$
negative_passages.docid: 77891
negative_passages.text: $(U\circ T)^{*} = T^{*}\circ U^{*}$
negative_passages.docid: 138342
negative_passages.text: Proof of a matrix exponential identity
negative_passages.docid: 26570
negative_passages.text: Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
negative_passages.docid: 33196
negative_passages.text: show that A(T) is a group under operation of composition of functions
negative_passages.docid: 24072
negative_passages.text: Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$
negative_passages.docid: 129732
negative_passages.text: What is $\operatorname{Aut}(\mathbb{Z}_2\oplus \mathbb{Z}_2)$?
negative_passages.docid: 25009
negative_passages.text: $\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
negative_passages.docid: 42809
negative_passages.text: How can we show that $(I-A)$ is invertible?
negative_passages.docid: 60598
negative_passages.text: Matrix $A^T A$ as sum of outer products
negative_passages.docid: 52354
negative_passages.text: Given a idempotent endomorphism, show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$
negative_passages.docid: 94077
negative_passages.text: Proof of linear independence of $e^{at}$
negative_passages.docid: 103839
negative_passages.text: Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$
negative_passages.docid: 128361
negative_passages.text: The kernel and range of the powers of a self-adjoint operator
negative_passages.docid: 90757
negative_passages.text: $T$ self-adjoint on Hilbert space $H$, $T^n=T^{n+1}$ will $T$ be an orthogonal projection operator?
negative_passages.docid: 51302
negative_passages.text: Show that the operator norm is submultiplicative
negative_passages.docid: 133756
negative_passages.text: Geometric interpretation of $\det(A^T) = \det(A)$