query: If $U \subset V$ are vectors spaces, then $V = U \oplus X$ for some vector space
positive_passages.docid: 18373
positive_passages.text: Can infinite-dimensional vector spaces be decomposed into direct sum of its subspaces?
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: 34768
negative_passages.text: Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$
negative_passages.docid: 4319
negative_passages.text: Vector space can be written as direct sum of subspaces
negative_passages.docid: 51397
negative_passages.text: If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$
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: 131993
negative_passages.text: Prove that vector space and dual space have same dimension
negative_passages.docid: 146339
negative_passages.text: Proof of -(-v)=v in a vector space
negative_passages.docid: 43504
negative_passages.text: Proving the existence of a complement subspace $U' \subset V$.
negative_passages.docid: 94405
negative_passages.text: Let $W$ be the set of all linear combinations of $v_1,...,v_n$. Then $W$ is a subspace of $V$.
negative_passages.docid: 118049
negative_passages.text: Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
negative_passages.docid: 137873
negative_passages.text: Problems that involves two facts in the definition of vector spaces
negative_passages.docid: 19396
negative_passages.text: For subspaces, if $N\subseteq M_1\cup\cdots\cup M_k$, then $N\subseteq M_i$ for some $i$?
negative_passages.docid: 66283
negative_passages.text: Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v
negative_passages.docid: 67191
negative_passages.text: When is the union of a family of subspaces of a vector space also a subspace?
negative_passages.docid: 123408
negative_passages.text: Why a subspace of a vector space is useful
negative_passages.docid: 28238
negative_passages.text: A question about long exact sequence of finite dim vector spaces