Data recorder example. A certain planar kinematics problem records the data set V using three components x,y,z. The working set S is a plane described by an ideal equation ax+by+cz=0. This plane is the hidden subspace of the physical application, obtained by a computation on the original data set V. More on vector spaces and subspaces: Detection of subspaces and data sets that are not subspaces. Theorems: Subspace criterion, Kernel theorem, Not a subspace theorem. Use of theorems 1,2 in section 4.2. Problem types in 4.1, 4.2. Example: Subspace Shortcut for the set S in R^3 defined by x+y+z=0. Avoid using the subspace criterion on S, by writing it as Ax=0, followed by applying the kernel theorem (thm 2 section 4.2). Subspace applications. When to use the kernel theorem. When to use the subspace criterion. When to use the not a subspace theorem. Problems 4.1,4.2.

Drill: The 8-property vector toolkit. Example: Prove zero times a vector is the zero vector. The kernel: Solutions of Ax=0. Find the kernel of the 2x2 matrix with 1 in the upper right corner and zeros elsewhere.

Vectors as packages of data items. Vectors are not arrows. Examples of vector packaging in applications. Fixed vectors. Gibbs motions. Physics i,j,k vectors. Arrows in engineering force diagrams. Functions, solutions of DE. Matrices, digital photos. Sequences, coefficients of Taylor and Fourier series. Hybrid packages. The toolkit of 8 properties. Subspaces. Data recorder example. Data conversion to fit physical models. Subspace criterion (Theorem 1, 4.2). Kernel theorem (Theorem 2, 4.2). Not a Subspace Theorem.

Example: c1 e^x+ c2 xe^{-x} = 2 e^x + 3 e^{-x} ==> c1=2, c2=3. Solutions of differential equations are vectors. Geometric tests One vector v1. Two vectors v1, v2. Algebraic tests. Rank test. Determinant test. Sampling test. Additional tests Wronskian test. Orthogonal vector test. Pivot theorem. Geometric tests. One or two vector independence. Geometry of dependence in dimensions 1,2,3.

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Additional Independence Tests Wronskian test. Orthogonal vector test. Pivot theorem [this lecture]. THEOREM: Pivot columns are independent and non-pivot columns are linear combinations of the pivot columns. THEOREM: rank(A)=rank(A^T). THEOREM: A set of nonzero pairwise orthogonal vectors is linearly independent. Basis. General solutions with a minimal number of terms. Definition: Basis == independence + span. Differential Equations: General solution and shortest answer. Pivot Theorem. Applications of the pivot theorem to find a largest set of independent vectors. Maximum set of independent vectors from a list.

ANNOUNCEMENT: Problem session 4.3, 4.4, 4.7 on Wed in WEB 104 and Thu in JTB 140. Solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26. Please refer to the chapter 4 problem notes.

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How to construct solutions to 4.3-18,24, 4.4-6,24 and 4.7-10,22,26. Questions answered on 4.3, 4.4, 4.7 problems. Survey of 4.3 problems. Illustration: How to do abstract independence arguments using vector packages, without looking inside the packages. Applications of the rank test and determinant test. How to use the pivot theorem to identify independent vectors from a list.

MAPLE LAB 2. [laptop projection] Solution to L2.2. Graphic in L2.3. Interpretation of graphics in L2.4. PROBLEMS. 3.6-40: adjugate details, how to get det(A)=107. Answer check: with(linalg):A:=matrix([[2,4,-3],[2,-3,-1],[-5,0,-3]]); inverse(A); det(A); adjoint(A); evalm(det(A)*inverse(A)); 3.6-60: Reading on induction. Required details. B_n = 2B_{n-1} - B_{n-2}, B_n = n+1 3.6-review: matrix A is 10x10 and has 92 ones. What's det(A)? ALGEBRAIC TESTS: mostly review Rank test. Determinant test. Sampling test. Wronskian test. Orthogonal vector test. Pivot theorem. FUNCTIONS. How to represent functions as graphs and as infinitely long column vectors. Rules for add and scalar multiply. Independence tests using functions as the vectors.. PROOFS. [slides] The pivot theorem. Algorithm 2, section 4.5. rank(A)=rank(A^T). Theorem 3, section 4.5. DIGITAL PHOTOS. Digital photos are matrices Photos are vectors == data packages Checkerboards and digital photos Matrix add and RGB separation, visualization Matrix scalar multiply, visualization BASIS. Definition of basis and span. Examples: Find a basis from a general solution formula. Bases and the pivot theorem. DIMENSION. THEOREM. Two bases for a vector space V must have the same number of vectors. Examples: Last Frame Algorithm: Basis for a linear system Ax=0. Last frame algorithm and the vector general solution. Basis of solutions to a homogeneous system of linear algebraic equations. Bases and partial derivatives of the general solution on the invented symbols t1, t2, ... DE Example: y = c1 e^x + c2 e^{-x} is the general solution. What's the basis? SOLUTION ATOMS and INDEPENDENCE. Def. atom=x^n(base atom) base atom = 1, exp(ax), cos(bx), sin(bx), exp(ax) cos(bx), exp(ax) sin(bx) "atom" abbreviates "solution atom of a linear differential equation" THEOREM. Atoms are independent. EXAMPLE. Show 1, x^2, x^9 are independent EXAMPLE. Show 1, x^2, x^(3/2) are independent [Wronskian test] PARTIAL FRACTION THEORY. Examples. top=x-1, bottom=(x+1)(x^2+1) top=x-1, bottom=(x+1)^2(x^2+1)^2 Maple assist with convert(top/bottom,parfrac,x); PROBLEM 4.7-26. How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x) Outline of the general theory used to solve linear differential equations. Order of a DE and the dimension of the solution space. Euler's theorem. Finding solution atoms for a basis.

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