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Answers to Exercises Linear Algebra Jim Hefferon ¡1¢ 3 ¡2¢ 1 ﬂ1 2ﬂ ﬂ3 1ﬂ ¡ ¢ x1 ¢ 3 ¡2¢ 1 ﬂx ¢1 2ﬂ ﬂx ¢3 1ﬂ ¡6¢ 8 ¡2¢ 1 ﬂ6 2ﬂ ﬂ8 1ﬂ R N f::: ﬂ :::g h:::i V;W;U ~v;w~ 0, 0V B;D En = h~e1; :::; ~eni ﬂ;– RepB(~v) Pn Mn£m [S] M `N V » W h;g H;G t;s T;S RepB;D(h) hi;j jTj R(h);N (h) R1(h);N1(h) Notation real numbers natural numbers: f0;1;2;:::g complex numbers set of ... such that ... sequence; like a set but order matters vector spaces vectors zero vector, zero vector of V bases standard basis for Rn basis vectors matrix representing the vector set of n-th degree polynomials set of n£m matrices span of the set S direct sum of subspaces isomorphic spaces homomorphisms, linear maps matrices transformations; maps from a space to itself square matrices matrix representing the map h matrix entry from row i, column j determinant of the matrix T rangespace and nullspace of the map h generalized rangespace and nullspace Lower case Greek alphabet name alpha beta gamma delta epsilon zeta eta theta character ﬁ ﬂ ° – † ‡ · µ name iota kappa lambda mu nu xi omicron pi character ¶ • ‚ „ ” » o … name rho sigma tau upsilon phi chi psi omega character ‰ ¾ ¿ À ` ´ ˆ ! Cover. This is Cramer’s Rule for the system x+2y = 6, 3x+y = 8. The size of the ﬁrst box is the determinant shown (the absolute value of the size is the area). The size of the second box is x times that, and equals the size of the ﬁnal box. Hence, x is the ﬁnal determinant divided by the ﬁrst determinant. These are answers to the exercises in Linear Algebra by J. Hefferon. Corrections or comments are very welcome, email to jimjoshua.smcvt.edu An answer labeled here as, for instance, One.II.3.4, matches the question numbered 4 from the ﬁrst chapter, second section, and third subsection. The Topics are numbered separately. Contents Chapter One: Linear Systems 3 Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Subsection One.I.2: Describing the Solution Set . . . . . . . . . . . . . . . . . . . . . . . 10 Subsection One.I.3: General = Particular + Homogeneous . . . . . . . . . . . . . . . . . 14 Subsection One.II.1: Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Subsection One.II.2: Length and Angle Measures . . . . . . . . . . . . . . . . . . . . . . 20 Subsection One.III.1: Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . . 25 Subsection One.III.2: Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter Two: Vector Spaces 36 Subsection Two.I.1: Deﬁnition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 37 Subsection Two.I.2: Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . . . 40 Subsection Two.II.1: Deﬁnition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 46 Subsection Two.III.1: Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Subsection Two.III.2: Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Subsection Two.III.3: Vector Spaces and Linear Systems . . . . . . . . . . . . . . . . . . 61 Subsection Two.III.4: Combining Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 66 Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter Three: Maps Between Spaces 74 Subsection Three.I.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . 75 Subsection Three.I.2: Dimension Characterizes Isomorphism . . . . . . . . . . . . . . . . 83 Subsection Three.II.1: Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Subsection Three.II.2: Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . . 90 Subsection Three.III.1: Representing Linear Maps with Matrices . . . . . . . . . . . . . 95 Subsection Three.III.2: Any Matrix Represents a Linear Map . . . . . . . . . . . . . . . 103 Subsection Three.IV.1: Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 107 Subsection Three.IV.2: Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 108 Subsection Three.IV.3: Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 112 Subsection Three.IV.4: Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Subsection Three.V.1: Changing Representations of Vectors . . . . . . . . . . . . . . . . 121 Subsection Three.V.2: Changing Map Representations . . . . . . . . . . . . . . . . . . . 124 Subsection Three.VI.1: Orthogonal Projection Into a Line . . . . . . . . . . . . . . . . . 128 Subsection Three.VI.2: Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . 131 Subsection Three.VI.3: Projection Into a Subspace . . . . . . . . . . . . . . . . . . . . . 137 Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Topic: Geometry of Linear Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Chapter Four: Determinants 158 Subsection Four.I.1: Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Subsection Four.I.2: Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 161 Subsection Four.I.3: The Permutation Expansion . . . . . . . . . . . . . . . . . . . . . . 164 Subsection Four.I.4: Determinants Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Subsection Four.II.1: Determinants as Size Functions . . . . . . . . . . . . . . . . . . . . 168 Subsection Four.III.1: Laplace’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 171 ... - tailieumienphi.vn
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