Final Exam Outline

1.1, 1.2 Scalars and vectors in Rn. The magnitude of a vector. Row and column vectors. Vector addition. Scalar multiplication. The zero vector. Vector subtraction. Distance in Rn. Unit vectors. Linear combinations. The dot product. The angle between two vectors. Orthogonality. The CBS and triangle inequalities. Theorem 1.5.

2.1, 2.2 Systems of linear equations. Consistent and inconsistent systems. The coefficient and augmented matrices. Elementary row operations. The echelon form of a matrix. Gaussian elimination. Equivalent matrices. The reduced row echelon form. Gauss-Jordan elimination. Leading and free variables. The rank of a matrix. The rank theorem. Homogeneous systems. Theorem 2.2.

2.3 The span of a set of vectors. Linear dependence. Linear dependence and independence. Determining n whether a set {~v1, . . . ,~vk} of vectors in R is linearly dependent or independent.

3.1, 3.2 Matrix operations. Scalar multiplication, matrix addition, matrix multiplication. The zero and identity matrices. Partitioned matrices. The outer product form. The standard unit vectors. The transpose of a matrix. A system of linear equations in matrix form. Algebraic properties of the transpose.

3.3 The inverse of a matrix. Invertible (nonsingular) matrices. The uniqueness of the solution to Ax = b when A is invertible. The inverse of a nonsingular 2 × 2 matrix. Elementary matrices and EROs. Equivalent conditions for invertibility. Using Gauss-Jordan elimination to invert a nonsingular matrix. Theorems 3.7, 3.9.

n n n 3.5 Subspaces of R . {0} and R as subspaces of R . The subspace span(v1,..., vk). The the row, column and null spaces of a matrix. Basis, dimension. Using Gaussian elimination to find a bases for subspaces of the form span(v1, . . . , vk) and for null(A). The nullity of a matrix. Characterizations of rank: rank(A) = dim(col(A)) = dim(row(A)). The rank theorem. The Fundmental Theorem of Invertible Matrices. Coordinates with respect to a basis. Theorems 3.23-3.26, 3.27 (parts), 3.28 and 3.29.

4.1, 4.3 Eigenvalues, eigenvectors and eigenspaces. The characteristic polynomial. Finding a basis for an eigenspace.

4.2 Determinants. Cofactor expansions. The Laplace expansion theorem. Determinants of triangular ma- trices. Properties of determinants. Cramer’s rule. The classical adjoint of a square matrix. Computing the inverse using the classical adjoint. Theorems 4.2, 4.3 (parts) 4.11 and 4.12.

4.4 Similar matrices. Diagonalization of matrices. Theorems 4.23-4.25.

5.1 Orthogonal sets in Rn. Orthogonal and orthonormal bases. Orthogonal matrices. Theorems 5.1-5.8.

5.2 The orthogonal complement W ⊥ of a subspace W . The orthogonal projection. Theorem 5.9, parts (a), (c) and (d). Theorem 5.11. (The Orthogonal Decomposition Theorem.)

5.3 The Gram-Schmidt procedure.

5.4 The eigenvalues and eigenvectors of real, symmetric matrices. Orthogonal diagonalization of real, symmetric matrices. Theorems 5.17-5.19.