## Applied linear algebra |

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algebraic applications approximate augmented matrix axioms basis vectors calculation Chapter characteristic polynomial chosen basis coefficient matrix column rank column vectors complex numbers conjugate constant constraints convex set corresponding deduce defined denotes dependent differential equations dimension dimensional vector space echelon form eigenvalues eigenvectors elementary matrices elements equations Ax Example expression Fourier series function Gaussian elimination given gives Hence Hint image space inner product space iteration kernel linear equations linear programming linear programming problem linear transformation linearly independent linearly independent vectors maximum method minimised mutually orthogonal non-singular non-zero nXn matrix obtain orthogonal matrix orthonormal basis P'AP pivotal polynomials of degree principal axes Proof Prove quadratic form quadric real number real polynomials real symmetric real vector space represented scalar product search directions Section set of linearly set of vectors shows simple simplex solution solving symmetric matrix Theorem unique vertex zero