query: Concise proof of $\operatorname{dim} V =\operatorname{dim\,ker}T+\operatorname{dim \, range}T$
positive_passages.docid: 24072
positive_passages.text: Intuitive explanation of why $\dim\operatorname{Im} T + \dim\operatorname{Ker} T = \dim V$
negative_passages.docid: 103380
negative_passages.text: Is $\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$ true?
negative_passages.docid: 77891
negative_passages.text: $(U\circ T)^{*} = T^{*}\circ U^{*}$
negative_passages.docid: 92510
negative_passages.text: Proving a vector space is infinite-dimensional
negative_passages.docid: 102371
negative_passages.text: Proving that two vector spaces of equal dimension are isomorphic
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: 62065
negative_passages.text: Prove that $v_1, \dots v_n$ is a basis of V.
negative_passages.docid: 41026
negative_passages.text: How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
negative_passages.docid: 62303
negative_passages.text: For $V$ vector space of dimension $n$ under $\mathbb C$ and $T: V \to V $ linear transformation , show $V= \ker T^n \oplus $ Im $T^n$
negative_passages.docid: 117131
negative_passages.text: Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$
negative_passages.docid: 15300
negative_passages.text: How to show that $(W^\bot)^\bot=W$ (in a finite dimensional vector space)
negative_passages.docid: 108742
negative_passages.text: Proving commutativity of addition for vector spaces
negative_passages.docid: 44281
negative_passages.text: Proof of sum of kernels of concatenated vector
negative_passages.docid: 63036
negative_passages.text: Proving that $\dim(\mathrm{span}({I_n,A,A^2,...})) \leq n$
negative_passages.docid: 11557
negative_passages.text: How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?
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: 94929
negative_passages.text: Proof that the all bases of a subspace V consist of the same number of vectors?