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Linear Matrices 43
Vector Spaces 144
Linear Transformations 176
Eigenvalues and Eigenvectors 210
augmented matrix axioms basis vectors called characteristic equation characteristic polynomial coefficient matrix cofactor column space column vectors coordinates defined det(A determinant diagonal matrix dimension distributive law echelon form eigenvalues eigenvector eigenvectors corresponding element elementary row operations entries Figure following system form a basis Form the matrix fourth row function given Gram-Schmidt process Hence homogeneous system identity matrix implies inner product space invertible matrix linear combination linear equations linear transformation linearly independent vectors m x n matrix of coefficients n x n matrix norm orthonormal basis Problem Solving Examples real numbers reduced row-echelon form Replace row row space row vectors scalar multiplication second row semi-magic square set of vectors Solving this system spanned square matrix standard basis subspace symmetric matrix system of equations system of linear THEOREM third row trivial solution unique solution variables vector space