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Linear Algebra 11 Course Material and time required Topic Days expected Unit 1: Prep and Review 1 day Equations of Lines Graphing Lines Algebra, Linear Equations Systems of equations 4 days Additional notes First week is intended to be a review week. To get students up to where I want them to be for the main course work. Section 2.1 in the textbook. Unit 2: Vectors 2 3 Vectors in R and R Vector notation 2 Days Vector equation of a line Other Definitions Vectors in Rn Vector Spaces 3 Days Subspaces Span and Linear Independence Standard Basis Length and Dot Product In R2 and R3 2 Days Definitions: Dot product, norm, Triangle Inequality, Unit Vectors Projection and Minimum Distance Projections 1 Day Minimum Distance 1 Day Cross Product and Volumes Cross Product Definition 1 Day Length of the cross product Volume Unit 3: Introduction to Matrices The Matrix Representation of a System Matrix Representation 1 Day Row operations Row echelon form 1 Day Which systems have solutions? 1 Day Short cuts and bad ideas Reduced Row Echelon Form Definition Do we even Care? Rank 1 Day Homogeneous Solutions Infinite solutions Section 1.1 Section 1.2 Section 1.3 Give a day of work in here for students to play around with all those definitions introduced. Section 1.4 Section 1.5 Work days should go in here. At least 2. Rest of section 2.1 Lots of examples Example 12 in the Text Section 2.2 Applications Span of Vectors Linear Independence Bases of Subspaces 1 Day 1 Day 1 Day Applications of Systems Linear Programming 1 Day Unit 4: Matrices 2 Matrix Operations Equality, Addition, Scalar 1 Day Multiplication Transpose Matrix Multiplication 2 Days Summation Notation Multiplication Facts Identity Matrix Matrix Mapping and Linear Maps Matrices are functions for Vectors! 1 Day Properties Specifics: Linear mappings 1 Day Linear Map = Matrix Map? Composition of Maps 1 Day Geometric Maps Rotations 1 Day Rank Theorem Solution Space Null Space Solution Set of Ax = b Section 2.3 These sections are JUST long enough to introduce and then do some examples. Add some time to practice all of section 2 in this space. Section 2.4 Section 3.1 Keep referencing back to normal Multiplication of numbers. (How multiplication is usually commutative, has a cancellation law, etc.) Section 3.2 Properties are the natural extension of normal functions Section 3.3 Short – For interest and time to give work time to students Section 3.4 1 Day Range of L and Column Space of A 1 Day Rowspace 2 Days Bases of Row/ColumnSpace Rank-Nullity Theorem Inverse Matrices What is that? 1 Day Procedure Some Facts: Summary Unit 5: Vector Spaces Spaces of Polynomials Polynomial addition subtraction <1 Day Properties (Familiar?) Examples Vector Spaces Abstract Definition End of Day with Sec. 4.1 Section 3.5 Section 4.1 Section 4.2 Examples of Vector Spaces 1 Day Subspaces Bases and Dimensions Section 4.3 Definitions 1 Day Obtaining a basis from a spanning set of vectors All bases have the same dimension 1 Day How to make a basis out of a LI set Theorem 4 and conclusion Unit 6: Determinants Finding Inverses Section 5.1 In terms of cofactors 1 Day 3x3 case 1 Day Solo work time in here General Case Row Operations Section 5.2 How Row operations change 1 Day the determinant Related to Invertibility 1 Day Product of matrices Matrix inverse by Cofactors Section 5.3 Definition of a cofactor matrix 1 Day A-1 in terms of cofactors Area, Volume, Determinant Section 5.4 How are area and determinant related? 1 Day Determinant and volume Unit 7: Eigenvalues and Eigenvectors, Diagonalization Eigenvalues and eigenvectors Section 6.1 What even are those? 1 Day Of a mapping? Of a matrix? Procedure to find them At least 2 days Lots of practice Diagonalization Section 6.2 Related to Eigenvalues/vectors D = PAP-1, how to find THE END!