query: Linear independence of images by $A$ of vectors whose span trivially intersects $\ker(A)$
positive_passages.docid: 142229
positive_passages.text: Prove that if $V \cap \text{Span}\{u_1, u_2,...,u_k\} = \{ 0\} $, then $Au_1, Au_2,...,Au_k$ are linearly independent
negative_passages.docid: 42495
negative_passages.text: Determine if the following sets of vectors are linearly independent
negative_passages.docid: 41347
negative_passages.text: The dimension of the intersection of two vector subspaces
negative_passages.docid: 106756
negative_passages.text: Union and Intersection of Sets $A_{i}$ and $B_{j}$
negative_passages.docid: 92774
negative_passages.text: Prove that a linear subspace of $C([0,1])$ is closed
negative_passages.docid: 145039
negative_passages.text: Algebraic independence over $\overline{\mathbb Q}$ and over $\mathbb Q$
negative_passages.docid: 30017
negative_passages.text: Orthogonality and linear independence
negative_passages.docid: 58774
negative_passages.text: There exist an infinite subset $S\subseteq\mathbb{R}^3$ such that any three vectors in $S$ are linearly independent.
negative_passages.docid: 53859
negative_passages.text: Continuous image of separable space
negative_passages.docid: 143517
negative_passages.text: Isomorphisms Between a Finite-Dimensional Vector Space and its Dual
negative_passages.docid: 139951
negative_passages.text: Direct Sum of vector subspaces
negative_passages.docid: 92534
negative_passages.text: Prove that any finite subset of a linearly independent set is linearly independent
negative_passages.docid: 55752
negative_passages.text: Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent
negative_passages.docid: 60063
negative_passages.text: Independence between random vectors
negative_passages.docid: 4717
negative_passages.text: The union of linear subspaces is not necessarily a subspace
negative_passages.docid: 69893
negative_passages.text: An inner product space and its proper closed subspace with trivial orthogonal complement
negative_passages.docid: 8291
negative_passages.text: If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?