query: Linear Algebra-Vector Spaces-Sub-Spaces
positive_passages.docid: 151940
positive_passages.text: Cancellation law for direct sums
negative_passages.docid: 146143
negative_passages.text: Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands
negative_passages.docid: 48113
negative_passages.text: Find a basis for the vector space of symmetric matrices with an order of $n \times n$
negative_passages.docid: 113261
negative_passages.text: How to find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is not isomorphic to $D?$
negative_passages.docid: 78604
negative_passages.text: How to find basis of intersection of 2 vector spaces with given basis?
negative_passages.docid: 4374
negative_passages.text: Geometric interpretation of a vector space and subspace?
negative_passages.docid: 120344
negative_passages.text: Are there nontrivial vector spaces with finitely many elements?
negative_passages.docid: 72077
negative_passages.text: ''Linear'' transformations between vector spaces over different fields .
negative_passages.docid: 113326
negative_passages.text: Proving orthogonality of vectors in a complex vector space
negative_passages.docid: 41520
negative_passages.text: Prove that $L(V,W)$ forms a vector space
negative_passages.docid: 43503
negative_passages.text: Bases of complex vector spaces and the axiom of choice
negative_passages.docid: 44710
negative_passages.text: Vector space of real vectors over field complex scalars.
negative_passages.docid: 81628
negative_passages.text: Why do we call a vector space in terms of vector space over some field?
negative_passages.docid: 110654
negative_passages.text: Vector Spaces and AC
negative_passages.docid: 52596
negative_passages.text: A basis for $k(X)$ regarded as a vector space over $k$
negative_passages.docid: 4151
negative_passages.text: Every subspace of a vector space has a complement
negative_passages.docid: 54217
negative_passages.text: Let $V$ be a vector space. Determine all linear transformations $T:V \rightarrow V$ such that $T=T^2$.