query: Does there exists a matrix A ∈ Mn(F) such that $I, A, A2 , ..., An$ are linearly independent in the vector space Mn(F).
positive_passages.docid: 143187
positive_passages.text: Matrix power series has linearly dependent terms
negative_passages.docid: 59803
negative_passages.text: Converse of Schur's Lemma in finite dimensional vector spaces
negative_passages.docid: 27478
negative_passages.text: Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
negative_passages.docid: 2287
negative_passages.text: If a set is linearly independent, a finite subset of that set is also linearly independent
negative_passages.docid: 86749
negative_passages.text: Prove that any set of vectors $S \subset \mathbb{R}^n$ containing the zero vector cannot be linearly independent
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: 64395
negative_passages.text: If $v_1,...,v_r$ are eigenvectors that correspond to distinct eigenvalues, then they are linearly independent.
negative_passages.docid: 108012
negative_passages.text: A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$
negative_passages.docid: 62058
negative_passages.text: How to prove $1$,$\sqrt{2},\sqrt{3}$ and $\sqrt{6}$ are linearly independent over $\mathbb{Q}$?
negative_passages.docid: 41611
negative_passages.text: Open and closed set in n by n matrices space.
negative_passages.docid: 57649
negative_passages.text: Prove that for every $f$, a bilinear form, there exists a basis ${v_1,...,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$
negative_passages.docid: 109498
negative_passages.text: Linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$
negative_passages.docid: 11620
negative_passages.text: Is it true that any matrix in $M_2(\mathbb R)$ is the sum of two squares?
negative_passages.docid: 119782
negative_passages.text: Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.
negative_passages.docid: 64353
negative_passages.text: Does the set of matrix commutators form a subspace?
negative_passages.docid: 62237
negative_passages.text: Characteristic polynomial equals minimal polynomial iff $x, Mx, \ldots, M^{n-1} x$ are linearly independent
negative_passages.docid: 154724
negative_passages.text: Every vector space has a basis