Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Contents<br />
Chapter One: <strong>Linear</strong> Systems 3<br />
Subsection One.I.1: Gauss’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
Subsection One.I.2: Describing the Solution Set . . . . . . . . . . . . . . . . . . . . . . . 10<br />
Subsection One.I.3: General = Particular + Homogeneous . . . . . . . . . . . . . . . . . 14<br />
Subsection One.II.1: Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
Subsection One.II.2: Length and Angle Measures . . . . . . . . . . . . . . . . . . . . . . 20<br />
Subsection One.III.1: Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . . . 25<br />
Subsection One.III.2: Row Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
Topic: Computer <strong>Algebra</strong> Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33<br />
Chapter Two: Vector Spaces 36<br />
Subsection Two.I.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
Subsection Two.I.2: Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . . . . 40<br />
Subsection Two.II.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 46<br />
Subsection Two.III.1: Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
Subsection Two.III.2: Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
Subsection Two.III.3: Vector Spaces and <strong>Linear</strong> Systems . . . . . . . . . . . . . . . . . . 61<br />
Subsection Two.III.4: Combining Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 66<br />
Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69<br />
Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70<br />
Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
Chapter Three: Maps Between Spaces 74<br />
Subsection Three.I.1: Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . 75<br />
Subsection Three.I.2: Dimension Characterizes Isomorphism . . . . . . . . . . . . . . . . 83<br />
Subsection Three.II.1: Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
Subsection Three.II.2: Rangespace and Nullspace . . . . . . . . . . . . . . . . . . . . . . 90<br />
Subsection Three.III.1: Representing <strong>Linear</strong> Maps with Matrices . . . . . . . . . . . . . 95<br />
Subsection Three.III.2: Any Matrix Represents a <strong>Linear</strong> Map . . . . . . . . . . . . . . . 103<br />
Subsection Three.IV.1: Sums and Scalar Products . . . . . . . . . . . . . . . . . . . . . 107<br />
Subsection Three.IV.2: Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 108<br />
Subsection Three.IV.3: Mechanics of Matrix Multiplication . . . . . . . . . . . . . . . . 112<br />
Subsection Three.IV.4: Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116<br />
Subsection Three.V.1: Changing Representations of Vectors . . . . . . . . . . . . . . . . 121<br />
Subsection Three.V.2: Changing Map Representations . . . . . . . . . . . . . . . . . . . 124<br />
Subsection Three.VI.1: Orthogonal Projection Into a Line . . . . . . . . . . . . . . . . . 128<br />
Subsection Three.VI.2: Gram-Schmidt Orthogonalization . . . . . . . . . . . . . . . . . 131<br />
Subsection Three.VI.3: Projection Into a Subspace . . . . . . . . . . . . . . . . . . . . . 137<br />
Topic: Line of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143<br />
Topic: Geometry of <strong>Linear</strong> Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147<br />
Topic: Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150<br />
Topic: Orthonormal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157<br />
Chapter Four: Determinants 158<br />
Subsection Four.I.1: Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159<br />
Subsection Four.I.2: Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 161<br />
Subsection Four.I.3: The Permutation Expansion . . . . . . . . . . . . . . . . . . . . . . 164<br />
Subsection Four.I.4: Determinants Exist . . . . . . . . . . . . . . . . . . . . . . . . . . . 166<br />
Subsection Four.II.1: Determinants as Size Functions . . . . . . . . . . . . . . . . . . . . 168<br />
Subsection Four.III.1: Laplace’s Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 171
Change language