query: If $T:V \to W$ is a linear transformation such that $T^2 = 0$, then $Im(T) \subseteq Ker(T)$?
positive_passages.docid: 16789
positive_passages.text: Prove that $T^2$ is the zero transformation if and only if $\mathrm{range}(T) \subseteq \ker(T)$.
negative_passages.docid: 77891
negative_passages.text: $(U\circ T)^{*} = T^{*}\circ U^{*}$
negative_passages.docid: 62077
negative_passages.text: Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
negative_passages.docid: 18379
negative_passages.text: Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$
negative_passages.docid: 78307
negative_passages.text: Under what conditions is $AA^T$ invertible?
negative_passages.docid: 65395
negative_passages.text: Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
negative_passages.docid: 61274
negative_passages.text: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
negative_passages.docid: 6202
negative_passages.text: $V$ be a vector space , $T:V \to V$ be a linear operator , then is $\ker (T) \cap R(T) \cong R(T)/R(T^2) $?
negative_passages.docid: 15175
negative_passages.text: When does a Möbius transformation map $\Im(z)>0$ to itself?
negative_passages.docid: 100107
negative_passages.text: How can I prove that a linear map $T:V\to V$ such that $T^2 = I$ is equal to the Identity map?
negative_passages.docid: 55388
negative_passages.text: Finding nilpotent and diagonalizable operators for a linear transformation $T$
negative_passages.docid: 23095
negative_passages.text: Prove that T is a linear transformation
negative_passages.docid: 109674
negative_passages.text: If $(I-T)^{-1}$ exists, can it always be written in a series representation?
negative_passages.docid: 37813
negative_passages.text: If $T\colon \mathbb R^n \to \mathbb R^n $ linear and $T^2 = kT$
negative_passages.docid: 45998
negative_passages.text: The range of $T^*$ is the orthogonal complement of $\ker(T)$
negative_passages.docid: 75059
negative_passages.text: Let $T:\mathbb{R}^n\to \mathbb{R}^n$ be a linear transformation...
negative_passages.docid: 2762
negative_passages.text: Mapping of zero vector under a linear transformation