query: why $(im(T^*))^{\perp}=KerT$
positive_passages.docid: 45998
positive_passages.text: The range of $T^*$ is the orthogonal complement of $\ker(T)$
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: 7384
negative_passages.text: How do I prove that $m(TE)=\left|\det(T)\right|m(E)$?
negative_passages.docid: 127000
negative_passages.text: When does $V = \ker(f) \oplus \operatorname{im}(f)$?
negative_passages.docid: 130253
negative_passages.text: Differentiating $\mbox{tr} (ABA^TC)$ w.r.t. $A$
negative_passages.docid: 146541
negative_passages.text: Why $\dim U+\dim U^\perp=\dim V$?
negative_passages.docid: 73406
negative_passages.text: Prove that $E_{\lambda}(T)=E_{\lambda^{-1}}(T^{-1})$ if $T$ is an invertible linear operator on a finite dimensional vector space $V$
negative_passages.docid: 103061
negative_passages.text: Why is it true that $\mathrm{adj}(A)A = \det(A) \cdot I$?
negative_passages.docid: 143608
negative_passages.text: Show the Trace$(B)^{2} \leq$ nTrace$(B^{T}B)$
negative_passages.docid: 89533
negative_passages.text: $\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n)) = \mathcal{K}(L^2(\mathbb{R}^m)) \otimes \mathcal{K}(L^2(\mathbb{R}^n))$?
negative_passages.docid: 143765
negative_passages.text: To Show that $\exp(tX)\exp(tY)= \exp(t(X+Y)) + (1/2)t^2[X, Y] +o(t^3)$
negative_passages.docid: 18282
negative_passages.text: Finding the minimal polynomial, the eigenvalues, and the characteristic polynomial of $T(A)=A+A^t$
negative_passages.docid: 149364
negative_passages.text: Characteristic polynomial of an inverse
negative_passages.docid: 78075
negative_passages.text: Why can I apply $HS^\dagger$ and then measure in the computational basis to measure $Y$?
negative_passages.docid: 15300
negative_passages.text: How to show that $(W^\bot)^\bot=W$ (in a finite dimensional vector space)
negative_passages.docid: 24585
negative_passages.text: Proving that $\mathbf{W}$+$\mathbf{W^{\perp}}$=$\mathbb{R^{n}}$
negative_passages.docid: 135514
negative_passages.text: $ \text{range } T' = (\ker T)^0$