Download Math 110 Review List

Math 110 Review List 1. The Geometry and Algebra of Vectors. Length and angle. The dot product. a. Definitions to know: A vector written as a linear combination of other vectors; dot product of two vectors; orthogonal vectors; length of a vector; the distance between two vectors; unit vector; the triangle inequality. b. Theorems to understand and be able to state: Theorem 1.1, 1.2, 1.3 and 1.5. c. Be able to: Perform all allowable vector operations; write a vector as a linear combination of other vectors; represent vectors (and linear combination of vectors) geometrically in the x -­‐ y plane; find the distance between two vectors; find a vector orthogonal to a given vector; find the projection of a vector onto another vector; find the angle between two vectors; find the area o€f a parallelogram. €
d. Relevant Sections: 1.1 and 1.2 2. Lines and Planes. The Cross Product. a. Definitions to know: normal vector to a plane; direction vector of a line b. Be able to: Given a variety of information -­‐ find the equation of a line or a plane (normal, general, vector and parametric form); find the shortest distance between a point and a plane, between a line and a plane, between two planes. c. Relevant Sections: 1.3 and Exploration. 3. Systems of Linear Equations a. Definitions to know: Linear equation; system of linear equations; equivalent systems of linear equations; solution to a system of linear equations; trivial and non-­‐trivial solutions; consistent and inconsistent system of equations; leading and free variables; row-­‐equivalent matrices; homogeneous system of linear equations; row-­‐reduced echelon form of a matrix. b. Theorems to understand and be able to state: 2.2 and 2.3. c. Be able to: Solve a system by Gaussian elimination, and by Gauss-­‐Jordan elimination; state your answer in parametric/vector form; interpret your answer geometrically. d. Relevant Sections: 2.1 and 2.2. 4. Spanning Sets and Linear Independence. a. Definitions to know: The span of a set of vectors; a linearly dependent and a linearly independent set of vectors; rank of a matrix. b. Theorems to understand and be able to state: 2.4, 2.5, 2.6, 2.7 and 2.8. c. Be able to: determine if a set of vectors is dependent or independent; obtain a subset of independent vectors from a given set of (possibly dependent) vectors; find a linearly independent set of vectors that spans a given set of vectors, determine if a given vector is in the span of a given set of vectors. 5.
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d. Relevant Sections: 2.3. Matrix Operations – algebra of matrices and the inverse of a matrix. a. Definitions to know: The identity matrix; the transpose of a matrix; a symmetric matrix; the inverse of a matrix, an elementary matrix. b. Theorems to understand and be able to state: 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13 and 3.14. c. Be able to: Perform all allowable algebraic operations on matrices (addition, scalar multiplication, matrix multiplication, finding the n -­‐th power of a matrix); write a matrix or its inverse as a product of elementary matrices; find the inverse of a matrix using Gauss-­‐Jordan elimination and using elementary matrices, €
d. Relevant Sections: 3.1, 3.2, and 3.3. Subspaces, Basis, Dimension and Rank. a. Definitions to know: Subspace of a given space; a basis; the dimension of space; the rank and the nullity of a matrix; the solution space of a given system; the row, the column and the null space of a matrix. b. Theorems to understand and be able to state: 3.19, 3.20, 3.21, 3.22, 3.23, 3.24, 3.25, 3.26, 3.27 and 3.29. c. Be able to: Determine if a vector belongs to a given subspace; determine if a set of vectors constitutes a basis for a given space; find a basis for a given space; find the rank of a matrix, find the solution space of a system; find the row, the column and the null space of a matrix. d. Relevant Sections: 3.5. Linear Transformations. a. Definitions to know: Linear transformation; a projection matrix; a rotation matrix; a permutation matrix; a reflection matrix, composite transformation; the inverse of a transformation; the identity transformation. b. Theorems to understand and be able to state: 3.31, 3.32 and 3.33. c. Be able to: find a matrix corresponding to a (composite) linear transformation; find the image of a vector under a linear transformation; find the pre-­‐image of a vector that has been transformed by a linear transformation; find the inverse of a transformation. d. Relevant Sections: 3.6. Determinants a. Definitions to know: Triangular matrix, a cofactor of a matrix, the adjoint of a matrix. b. Theorems to understand and be able to state: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12 and Lemma 4.14. c. Be able to: find the determinant of a matrix using the cofactor (by any row/column) method; find the determinant by row reducing to a triangular matrix; find the determinant of A given A is a product matrix of some given k matrices; find the inverse of a matrix using the adjoint method; Cramer’s rule for solving a system of linear equations. d. Relevant Sections: 4.2. Eigenvalues and Eigenvectors. a. Definitions to know: An eigenvalue, and eigenvector, the characteristic polynomial and characteristic equation; algebraic and geometric multiplicity of eigenvalues; eigenspace corresponding to a given eigenvalue; a basis for an eigenspace. b. Theorems to be able to state: 4.15, 4.16, 4.17 4.18 and 4.20. c. Be able to: find eigenvalues (real or complex numbers) and corresponding eigenvectors/ eigenspace. d. Relevant Sections: 4.1 and 4.3. 10. Similarity and Diagonalization a. Definitions to know: similar matrices; a diagonalizable matrix, a similarity invariant property. b. Theorems to understand and be able to state: 4.21, 4.22, 4.23, 4.25, Lemma 4.26 and Theorem 4.27. c. Be able to: Find a matrix P that diagonalizes a given matrix A; find the k -­‐
th power of a given matrix A by firstly writing it in PDP −1 form. d. Relevant Sections: 4.4 11. Orthogonality, orthogonal complements and orthogonal projections. €
a. Definitions to know: An orthogonal and an orthonormal set of vectors, a €
vector orthogonal to a given space; the orthogonal projection of a vector v onto a given space W ; the component of a vector v orthogonal to a given space W , orthogonal complement of a given space; an orthogonal matrix; the four fundamental spaces of a given matrix; an orthogonal and an orthonormal basis. €
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b. Theorems to understand and be able to state: 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, €
5.7, 5.8, 5.9, 5.10, 5.11, 5.13 and 5.14. c. Be able to: Find the projection of a vector onto a given space; find the component of a vector v orthogonal to a given space; find an orthogonal complement of a given space; find the orthogonal complement of the null space of a given matrix, find the orthogonal complement of the null space of the transpose of a given matrix; for a given matrix find the orthogonal €
complement of the row space and that of the column space; using the Gram-­‐Schmidt process to find an orthogonal and an orthonormal basis for a given space. d. Relevant Sections: 5.1, 5.2 and 5.3 (omit the QR factorization) 12. Orthogonal diagonalization of Symmetric Matrices a. Definitions to know: an orthogonally diagonalizable matrix. b. Theorems to understand and be able to state: 5.17, 5.18, 5.19 and 5.20. c. Be able to: write a given symmetric matrix A as A = PDP −1 . d. Relevant Sections: 5.3 13. Complex Numbers a. Definitions to know: Given a complex number z , define Re(z) and €
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Im(z) ; modulus of z ; conjugate of z ; polar and exponential form of z ; arg(z) , Arg(z) ; €
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b. Theorems to be able to state: De Moivre’s Theorem and Euler’s Formula. c. Be able to: do addition, subtraction, multiplication, division (in ordered pair form); convert a complex number from ordered pair form to polar or exponential form and vice-­‐versa; take the n -­‐th power and n -­‐th root of a complex number (in exponential or in polar form). d. Relevant Sections: Appendix C. €
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