Review for Second Midterm of Math 21b, November 28, 2007(1) Gram-Schmidt process of constructing orthonormal vectors from linearlyindependent vectors.(2) QR decomposition of an n × m matrix A in the form QR, where Q isan n × m matrix whose column vectors are orthonormal and R is an m × mmatrix which is upper triangular with positive diagonal entries.(3) Orthogonal transformations as length-preserving and orthogonality-preservingtransformations. Orthonormal set of vectors and orthonormal basis. Pythagoreantheorem. Cauchy-Schwarz inequality. Angle between vectors.(4) Transpose of a matrix. Product of transposes of matrices and inverse ofthe transpose of a matrix. Symmetric and skew-symmetric matrices. Innerproduct of two vectors as the matrix product of the transpose of a vectorand the other vector. The kernel of the transpose of a matrix as orthogonalcomplement of the image of the matrix.(5) Formula for the orthogonal projection onto a subspace spanned by or-thonormal vectors. Formula for the orthogonal projection onto a subspacespanned by linearly independent vectors. Formula for the least-squares solu-tion of a (possibly inconsistent) system of linear equations.(6) Determinant defined by induction and expansion down the first column.Determinant computed by expansion down any column and across any row.Formula for the determinant of a 2 × 2 matrix. Formula for the determinantof a 3 × 3 matrix. Formula for the determinant of an upper or lower trian-gular matrix. Formula for the determinant of an upper or lower triangularpartitioned matrix with square matrices as diagonal entries.(7) Effect of Gauss-Jordan row operations on a determinant. Computationof determinant by Gaussian elimination. Determinant of the transpose of amatrix. Determinant of a product of matrices. Determinant of the inverseof a matrix. Determinants of similar matrices.(8) Characteristic equation of a square matrix. Trace of a matrix. Eigen-values, eigenvectors, eigenspaces and their computations. Eigenbasis anddiagonalization. Algebraic and geometric multiplicities and their relation.1Diagonalization of a matrix whose eigenvalues are all real and distinct. Eigen-basis and diagonalization for a linear transformation of a linear space (withdomain and codomain of the linear transformation both equal to the samelinear space).(9) Discrete linear dynamical systems. Closed formula for the state vector ofthe system by using diagonalization with respect to an eigenbasis. Discretetrajectories and phase portraits.(10) Complex eigenvalues. Any real 2 × 2 matrix with non-real eigenvaluesis similar to a rotation-scaling matrix. Formula for such a similarity.(11) For a dynamical system defined by a real 2 × 2 matrix the zero state isasymptotically stable if and only if the modulus of all the complex eigenvaluesis less than 1.The Second Midterm covers up to and including Section 7.6of Bretscher’s book on Linear Algebra with Applications,except Sections 4.2, 4.3, 5.5, and
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