Chapter 5  Independence and Basis in Rm

• Section 5.1, pages 221-223: 1, 3, 7, 13, 15, 17, 19, 23, 25, 28, 29, 30, 31.  Key ideas: linear combinations, linear independence, linear dependence, elementary columns/vectors, testing for LI/LD
• Section 5.2, pages 226-227: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 27.  Key ideas: subspace, solutions space for a matrix, span of a set of vectors, finding spanning sets
• Section 5.3, pages 234-237: 1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 30.  Key Ideas: basis for a subspace, dimension, dimension of solution space, determining whether or not a set of vectors is a basis for a given subspace
• Section 5.4, pages 240-241: 1, 3, 7, 9, 13, 15, 25.  Key Ideas: column space, column rank, row space, row rank, rank theorem

Chapter 6  Vector Spaces

• Section 6.1, pages 258-259: 1, 3, 5, 7, 9, 11, 17, 19.  Key Ideas: axioms for vector spaces, subspaces, examples like matrices, polynomials, complex numbers, real-valued functions
• Section 6.2, pages 266-267: all odd.  Key Ideas: definition of linear independence for vector spaces, spanning set, basis, dimension
• Section 6.3, pages 271-273: 1, 3, 5, 7, 9, 13, 15, 19, 21.  Key Ideas: coordinate vectors, using coordinate vectors to determine if LI, spans
• Section 6.4, pages 283-285: 1, 3, 7, 9, 11, 17, 19, 21, 23.  Key Ideas: inner product, Cauchy-Schwarz inequality, orthogonal sets and orthogonal bases, Gram-Schmidt
• Section 6.8, pages 311: 1, 2, 3, 4.  Key Ideas: using vectors and orthogonal projections to make fractals, the Koch curve

Chapter 7  Linear Transformations

• Section 7.1, pages 328-329: 1,3,5,7,9,11,13,17,21, 25.  Key ideas: definition of a linear transformation, the zero transformation, the identity transformation, rotation, reflection, knowing what happens on a spanning set/basis is enough to know what happens everywhere.
• Section 7.2, pages 338-339: 1,3,5,7,9,11,13,17,27.  Key Ideas: The algebra of linear transformations (sum, difference, negative, scalar multiple, product and powers), be able to use/apply any of the items in Theorem 3, the matrix of a linear transformation, Theorems 5 and 6 for associated matrices
• Section 7.3, pages 347-348: 1,3,5,7,9,11,13,15,19,21,27,29,31.  Key Ideas: the kernel and image of a linear transformation, one-to-one, onto, nullity+rank=dimV
• Section 7.7, pages 377-378: 1, 2, 3.  Key Ideas: Contractive affine maps, iterated function systems (IFS)

Chapter 8  Eigenvalues and Eigenvectors

• Section 8.1, pages 389-390: 1, 3, 5, 7, 9, 11, 21, 23.  Key Ideas: definition of eigenvalue and eigenvector for a general linear transformation, eigenspace
• Section 8.2, pages 398-3400: 1, 3, 5, 13, 17, 19, 31.  Key Ideas: similar matrices (see notes), what it means for a matrix to be diagonalizable, procedure for diagonalizing, conditions for being diagonalizable, symmetric matrices, orthogonal matrices, orthogonally diagonalizable, condition for being orthogonally diagonalizable

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Last updated April 13, 2015