query: Linear Transformation from $\mathbb R^n$ to $\mathbb R^m$: image of $1$ dimensional subspace has dimension $1$ or $0$
positive_passages.docid: 18921
positive_passages.text: Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional
negative_passages.docid: 75829
negative_passages.text: Dimension of open subsets of $R^n$
negative_passages.docid: 140225
negative_passages.text: Linear transformation maps the first quadrant to a closed set
negative_passages.docid: 102284
negative_passages.text: $GL(n,\mathbb R)/SL(n,\mathbb R)$ is isomophic to $\mathbb R^\times$
negative_passages.docid: 125073
negative_passages.text: Finite-dimensional subspaces of normed vector spaces are direct summands
negative_passages.docid: 73690
negative_passages.text: If $\mathbb R^3\setminus V$ connected where $V$ is the subspace generated by $\{(1,1,1),(0,1,1)\} $
negative_passages.docid: 151425
negative_passages.text: Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?
negative_passages.docid: 15909
negative_passages.text: Bijection from $\mathbb R$ to $\mathbb {R^N}$
negative_passages.docid: 97049
negative_passages.text: If $v_1,...,v_m$ are linearly independent, then the span $v_1+w,...,v_m+w$ has dimension $\ge m-1$
negative_passages.docid: 53364
negative_passages.text: dim$(V)$ = $n$, dim$(W)$ = $m$ $\implies$ dim($L(V,W)$) = $nm$
negative_passages.docid: 38516
negative_passages.text: Show a subset is subspace of a vectorspace and find the dimension
negative_passages.docid: 117152
negative_passages.text: Preimage of codimension one subspace of the quotient $g/D^1 g$
negative_passages.docid: 88728
negative_passages.text: Dimensions of vector subspaces in a direct sum are additive
negative_passages.docid: 1799
negative_passages.text: How many k-dimensional subspaces there are in n-dimensional vector space over $\mathbb F_p$?
negative_passages.docid: 60684
negative_passages.text: Determine whether the set $W$ is a subspace in $\mathbb{R}^{3}$
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: 134065
negative_passages.text: A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.