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...Notes on mathematics peeyush chandra a k lal v raghavendra g santhanam supported by grant from mhrd contents i linear algebra matrices denition of matrix special operations multiplication some more submatrix block over complex numbers system equations introduction and solution method asolution row equivalent systems gauss elimination reduced echelon form jordan elementary rank existence ax b example main theorem exercises invertible inverse conditions for invertibility determinant adjoint cramer s rule miscellaneous finite dimensional vector spaces examples subspaces combinations independence bases important results ordered transformations denitions basic properties transformation nullity similarity inner product gram schmidt orthogonalisation process orthogonal projections applications the projection eigenvalues eigenvectors diagonalization diagonalizable sylvester law inertia ii ordinary dierential equation preliminaries separable reducible to exact integrating factors remarks initia...