All about the ⊥ with its applications in the linear statistical models

For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

Matrix Properties, Fundamental Spaces, Orthogonality

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an M × N matrix is associated with four fundamental spaces. The column space is the set of all M-vectors that are linear combinations of the columns. If the matrix has M independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left null space, the set of all M-vectors that the matrix’s s transpose maps to the zero N-vector. The row space is the set of all N-vectors that are linear combinations of the rows. If the matrix has N independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the null space, the set of all N-vectors that the matrix maps to the zero M-vector.

Viewspapers: The Malay press of the 1930s

There was a tremendous acceleration in newspaper publishing between 1930 and 1941 despite the Great Depression. The Malay press began to evolve into a site for discussing and debating the circumstances of Malay life in the 1930s. Rather than news, opinions, commentaries, leading articles and editorials made up the bulk of column space in Malay newspapers and magazines of the 1930s. It was a ‘viewspaper’ rather than a newspaper. New forms of public-opinion making like the editorial, increased participation in the media through letters to the editor and contributors' articles, public readings of newspapers, and the extension of newspapers into classrooms meant that a broader cross-section of Malays were able to access debates and discussions on issues of the day and raises new questions about public life in Malaya among Malays.