prompter: Is the set of all invertible matrices a linear subspace of the set of all square matrices over reals?
assistant: No, it is not. The set of all invertible matrices is not stable under matrix addition.
Here is a trivial counterargument: let M be an invertible matrix. Then -M is invertible, because det(-M) = -det(M). Now, the sum of M and -M is a null matrix, which is clearly not invertible.
prompter: Also see:
"""
Linear combination
Span
Basis
Dimension
Null space
Column space
Row space
Orthogonal complement
Orthonormal basis
Orthogonal projection
Orthogonal subspace
Orthogonal transformation
Inner product space
Orthogonal vector
Orthogonal matrix
Orthogonal diagonalization
Eigenvectors
Eigenvalues
Characteristic polynomial
Trace
Determinant
Matrix representation
Linear operator
Linear function
Linear mapping
Rank
Nullity
RREF (Reduced Row Echelon Form)
Gaussian elimination
LU decomposition
QR decomposition
Gram-Schmidt process
SVD (Singular Value Decomposition)
PCA (Principal Component Analysis)
Vector space
Affine space
Linear independence
Linear dependence
Independence matrix
Dependence matrix
Coordinate vector
Coordinate transformation
Change of basis
Matrix of a linear transformation
Image
Kernel
Range
Rank-nullity theorem
Trace operator
Transpose
Adjoint
Dual space
Dual basis
Orthogonal dual basis
Orthogonal projection operator
Orthogonal complement subspace
Orthogonal diagonalization theorem
Spectral theorem
Jordan form
Diagonal matrix
Triangular matrix
Symmetric matrix
Skew-symmetric matrix
Hermitian matrix
Skew-Hermitian matrix
Positive definite matrix
Positive semi-definite matrix
Negative definite matrix
Negative semi-definite matrix
Orthogonal diagonalization of symmetric matrices
Orthogonal diagonalization of Hermitian matrices
Unitary matrix
Unitary diagonalization
Orthogonal matrices and rotations
Orthogonal matrices and reflections
Orthogonal matrices and orthogonal transformations
Orthogonal matrices and isometries
Orthogonal matrices and orthonormal bases
Orthogonal matrices and orthogonal complements
Orthogonal matrices and orthogonal projections
Orthogonal matrices and orthogonal diagonalization
Orthogonal matrices and the Gram-Schmidt process
Orthogonal matrices and the QR decomposition
Orthogonal matrices and the SVD
Orthogonal matrices and the PCA
Orthogonal matrices and linear transformations
Orthogonal matrices and linear mappings
Orthogonal matrices and linear operators
Orthogonal matrices and linear functions
Orthogonal matrices and linear subspaces
Orthogonal matrices and vector spaces
Orthogonal matrices and affine spaces
Orthogonal matrices and linear independence
Orthogonal matrices and linear Simulations
Orthogonal matrices and linear combinations
Orthogonal matrices and eigenvectors
Orthogonal matrices and eigenvalues
Orthogonal matrices and the spectral theorem
Orthogonal matrices and the Jordan form
Orthogonal matrices and diagonal matrices
Orthogonal matrices and triangular matrices
Orthogonal matrices and the trace
Orthogonal matrices and the determinant
Orthogonal matrices and the transpose
Orthogonal matrices and the adjoint
Orthogonal matrices and the dual space
Orthogonal matrices and the dual basis
Orthogonal matrices and orthogonal dual basis
Orthogonal matrices and orthogonal projection operators
Orthogonal matrices and orthogonal complement subspaces
"""
assistant: Thank you for sharing this long list with me - but I'm unsure what you expect me to do. What further questions do you have that would allow me to support you accordingly?
prompter: Sorry for the list above. Can you explain the basic of Graph theory and how it is connected to programming?
assistant: Graph theory is a branch of maths that studies graphs, that are mathematical structures used to model relationships between objects. In computer science, graph theory is applied in several areas such as algorithms, data structures, and databases. Graphs are data structures which consist of a node (data item which can contain information) and edges (the connections between nodes which can be directed and have values).

Graph theory algorithms are used in computer science for various purposes, such as: path finding, analysing the flow of resources or information in a network, analysing relationships between objects in a dataset.

In programming, graph theory algorithms can be implemented in several ways, including adjacency matrices, adjacency lists, and edge lists. Graphs can be stored and manipulated using data structures such as arrays, linked lists, and trees. Many programming languages have libraries and packages that provide efficient implementations of graph algorithms, making it easy for developers to use graph theory in their applications.