query: $ |T z|= |z| \,\,\forall z\in \mathbb{C} \implies ab=0 $
positive_passages.docid: 145104
positive_passages.text: $T:$ $\mathbb{C}$ $\to$ $\mathbb{C}$ by $T(z)=\lambda z + \mu \bar{z}$ for every $z \in \mathbb{C}$
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: 13794
negative_passages.text: $|z_{1} + z_{2}| = |z_{1} - z_{2}| \implies z_{1}/z_{2}$ is Imaginary
negative_passages.docid: 143354
negative_passages.text: Prove that there exists $c\in\mathbb{R}$ such that $f'(c)=f(c)$
negative_passages.docid: 73773
negative_passages.text: $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.
negative_passages.docid: 67717
negative_passages.text: $A \oplus B = A \oplus C$ imply $B = C$?
negative_passages.docid: 83337
negative_passages.text: If $A ∈M_n(C)$ is such that $(Ax,x) = 0$ for all $x ∈C^n$, then $A = 0$
negative_passages.docid: 84313
negative_passages.text: $|f(z)|\le 1-|z|\forall z\in D$, we need to show $f\equiv 0$
negative_passages.docid: 139934
negative_passages.text: In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?
negative_passages.docid: 124320
negative_passages.text: $a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
negative_passages.docid: 83595
negative_passages.text: Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
negative_passages.docid: 83475
negative_passages.text: Prove Or Disprove: tr(AB)=tr(A)*tr(B)
negative_passages.docid: 18971
negative_passages.text: Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
negative_passages.docid: 55673
negative_passages.text: Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
negative_passages.docid: 85032
negative_passages.text: $Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
negative_passages.docid: 73869
negative_passages.text: $\mathcal{C}_1 \subseteq \mathcal{C}_2 \implies \sigma( \mathcal{C}_1) \subseteq \sigma( \mathcal{C}_2) $
negative_passages.docid: 50266
negative_passages.text: In the following figure prove that: $AK||BC$