query: Do the columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$?
positive_passages.docid: 147797
positive_passages.text: If $A$ is an invertible matrix, then the column span of $A$ equals $F^{n\times1}$
negative_passages.docid: 8557
negative_passages.text: Invertible Matrices are dense
negative_passages.docid: 64644
negative_passages.text: An $r\times r$ submatrix of independent rows and independent columns is invertible (Michael Artin's book).
negative_passages.docid: 115582
negative_passages.text: Prove that if a product$ AB$ of $n\times n$ matrices is invertible, so are the factors $A$ and $B$.
negative_passages.docid: 33355
negative_passages.text: Proving that matrix in equation is invertible
negative_passages.docid: 43282
negative_passages.text: Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?
negative_passages.docid: 112939
negative_passages.text: Find a basis for the subspace of $\mathbb{R}^4$ spanned by the given vectors.
negative_passages.docid: 33559
negative_passages.text: Product or sum of invertible matrix give an invertible matrix?
negative_passages.docid: 109217
negative_passages.text: Invertibility of the Product of Two Matrices
negative_passages.docid: 28402
negative_passages.text: Number of bases of an n-dimensional vector space over q-element field.
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: 120766
negative_passages.text: Prove ${y_n}$ is a basis for a Hilbert space $H$.
negative_passages.docid: 52040
negative_passages.text: "Inverse" of an $n\times k$ matrix.
negative_passages.docid: 64353
negative_passages.text: Does the set of matrix commutators form a subspace?
negative_passages.docid: 100142
negative_passages.text: Is the Pauli group for $n$-qubits a basis for $\mathbb{C}^{2^n\times 2^n}$?
negative_passages.docid: 63475
negative_passages.text: Is there any connection between a matrix being invertible and being diagonalizable?
negative_passages.docid: 144145
negative_passages.text: Symmetric $3\times 3$ Matrices - Base matrix