query: The working of $\dim(\ker(X))$ in a square matrix
positive_passages.docid: 22149
positive_passages.text: How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
negative_passages.docid: 96158
negative_passages.text: What's the associated matrix of this linear operator?
negative_passages.docid: 32064
negative_passages.text: Zero-dimensional matrices
negative_passages.docid: 111031
negative_passages.text: Matrix rank and linear independence of functions
negative_passages.docid: 104258
negative_passages.text: Proof Verification: $\dim\mathrm{Im} T+\dim\ker T=\dim V$
negative_passages.docid: 105141
negative_passages.text: $\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$
negative_passages.docid: 2109
negative_passages.text: Max dimension of a subspace of singular $n\times n$ matrices
negative_passages.docid: 121047
negative_passages.text: Proof: Sum of dimension of orthogonal complement and vector subspace
negative_passages.docid: 146541
negative_passages.text: Why $\dim U+\dim U^\perp=\dim V$?
negative_passages.docid: 11620
negative_passages.text: Is it true that any matrix in $M_2(\mathbb R)$ is the sum of two squares?
negative_passages.docid: 70606
negative_passages.text: Show that the LU decomposition of matrices of the form $\left[\begin{smallmatrix}0& x\\0 & y\end{smallmatrix}\right]$ is not unique
negative_passages.docid: 64353
negative_passages.text: Does the set of matrix commutators form a subspace?
negative_passages.docid: 101631
negative_passages.text: Determinant of a finite-dimensional matrix in terms of trace
negative_passages.docid: 10083
negative_passages.text: Determinant of matrix times a constant.
negative_passages.docid: 53661
negative_passages.text: Intuition surrounding units in $R[x]$
negative_passages.docid: 150720
negative_passages.text: Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
negative_passages.docid: 137209
negative_passages.text: Finding the Matrix of a Linear Operator Satisfying a Polynomial