query: Let $S$ be a subspace of an n-dimensional vector space $V$. Prove that $\operatorname{dim}(S)\leq\ n$.
positive_passages.docid: 150720
positive_passages.text: Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$
negative_passages.docid: 122853
negative_passages.text: Show that $\operatorname{rank}(A) = \operatorname{rank}(B)$
negative_passages.docid: 92291
negative_passages.text: $U_1, U_2, ..., U_k$ subspaces of linear space $X$ with same dimensions, prove $X=U_1\oplus V=U_2\oplus V=...=U_k\oplus V$
negative_passages.docid: 106487
negative_passages.text: Let $S$ and $W$ be subsets of a vector space $V$. Show that if $S$ is a subset of $W$, then $\mathrm{span}(S)$ is a subspace of $\mathrm{span}(W)$
negative_passages.docid: 53261
negative_passages.text: Show that Pn is an (n+1)-dimensional subspace
negative_passages.docid: 94929
negative_passages.text: Proof that the all bases of a subspace V consist of the same number of vectors?
negative_passages.docid: 124786
negative_passages.text: Prove that V contains a subspace of each dimension
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: 133087
negative_passages.text: Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.
negative_passages.docid: 74623
negative_passages.text: Associative rule of subspaces and vector spaces: $(U+V) \cap W = U + (V \cap W)$
negative_passages.docid: 141794
negative_passages.text: Invariant subspaces of specific dimension
negative_passages.docid: 18921
negative_passages.text: Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional
negative_passages.docid: 74764
negative_passages.text: Dimension for a closed subspace of $C[0,1]$.
negative_passages.docid: 60684
negative_passages.text: Determine whether the set $W$ is a subspace in $\mathbb{R}^{3}$
negative_passages.docid: 117688
negative_passages.text: Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
negative_passages.docid: 70475
negative_passages.text: Proving a basis for inner product space V when $||e_j-v_j||< \frac{1}{\sqrt{n}}$.
negative_passages.docid: 28565
negative_passages.text: Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$