Vectors can be added or subtracted by adding or subtracting their corresponding components.

The dot product of two vectors is a scalar value that represents the magnitude of their projection onto each other.

Gaussian elimination is a widely used algorithm for solving systems of linear equations by systematically reducing the system to an upper triangular form.

The determinant of a matrix is a scalar value that is used to characterize the matrix's properties, such as invertibility.

QR decomposition is commonly used for solving least squares problems because it allows for the efficient computation of the solution.

The Gram-Schmidt process is used to orthogonalize a set of vectors, meaning it transforms them into a set of mutually perpendicular vectors.

Power iteration is a widely used algorithm for finding the dominant eigenvalue and eigenvector of a matrix.

Singular value decomposition (SVD) factorizes a matrix into a product of three matrices, revealing important information about the matrix's structure and properties.

Strassen's algorithm is a widely used algorithm for matrix multiplication that is known for its efficiency, particularly for large matrices.

The Cholesky decomposition factorizes a symmetric positive definite matrix into a product of two triangular matrices, which is useful for solving systems of linear equations and other numerical computations.

Gaussian elimination can be used to find the rank of a matrix by reducing it to an echelon form and counting the number of nonzero rows.

The QR algorithm is an iterative method for finding the eigenvalues and eigenvectors of a matrix.

The conjugate gradient method is a widely used algorithm for solving sparse systems of linear equations, particularly when the matrix is symmetric positive definite.

The Lanczos algorithm is an iterative method for finding the eigenvalues and eigenvectors of a matrix, particularly for large sparse matrices.