query: Vector Spaces and Linear Transformations ($T^2 = 0 \iff R(T) \subseteq N(T)$).
positive_passages.docid: 149907
positive_passages.text: $T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
negative_passages.docid: 143517
negative_passages.text: Isomorphisms Between a Finite-Dimensional Vector Space and its Dual
negative_passages.docid: 91022
negative_passages.text: How do I prove that a subspace of a vector space $X$ is the null space of some linear functional on $X$?
negative_passages.docid: 118996
negative_passages.text: Prove that $T$ is an orthogonal projection
negative_passages.docid: 64156
negative_passages.text: Why an eigenspace is a linear subspace, if the zero vector is not an eigenvector?
negative_passages.docid: 123505
negative_passages.text: how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?
negative_passages.docid: 24657
negative_passages.text: Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.
negative_passages.docid: 98824
negative_passages.text: linear transformation $T$ such that $TS = ST$
negative_passages.docid: 59803
negative_passages.text: Converse of Schur's Lemma in finite dimensional vector spaces
negative_passages.docid: 99300
negative_passages.text: Prove if $W \subset V$ is a subspace such that $T(W)\subset W$, then $V/W$ is finite-dimensional.
negative_passages.docid: 154625
negative_passages.text: $\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$
negative_passages.docid: 24004
negative_passages.text: General form of a linear transformation
negative_passages.docid: 121636
negative_passages.text: General linear group of a vector space
negative_passages.docid: 112231
negative_passages.text: Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set
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: 88013
negative_passages.text: injective linear transformation basis.
negative_passages.docid: 18921
negative_passages.text: Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional