Edwards-Penney, sections 4.4, 4.5, 4.6, 4.7, 5.1 The textbook topics, definitions and theorems

Edwards-Penney 4.1, 4.2, 4.3, 4.4 (5.6 K, txt, 19 Dec 2013)

Edwards-Penney 4.5, 4.6, 4.7 (7.0 K, txt, 19 Dec 2013)

Edwards-Penney 5.1, 5.2, 5.3, 5.4 (15.6 K, txt, 19 Dec 2013)

Sections 4.1, 4.3 and some part of 4.7.

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.

Data recorder example. Data conversion to fit physical models. Subspace criterion (Theorem 1, 4.2). Kernel theorem (Theorem 2, 4.2). Span Theorem (Theorem 1, 4.3) Not a Subspace Theorem (Theorem 1, 4.2, backwards)

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: Review One vector v1. Two vectors v1, v2. Three vectors v1, v2, v3. Abstract vector space tests: Review One vector v1. Two vectors v1, v2. Algebraic tests: Review Given vectors v1, v2, ... fill them into the columns of matrix A. Rank test: Independent if and only if rank(A)=# columns of A Determinant test: Independent if and only if |A| is nonzero.

Pivot theorem. Additional tests Sampling test. Wronskian test. Orthogonal vector test. Web References: Vector space, subspace, independence tests (168.4 K, pdf, 17 Feb 2014)Slides: The pivot theorem and applications (189.2 K, pdf, 03 Mar 2012)Slides: Orthogonality (124.8 K, pdf, 03 Mar 2012)Slides: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)Manuscript: Rank, nullity and elimination (156.3 K, pdf, 21 Dec 2012)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012)Slides

THEOREM: Pivot columns are independent and non-pivot columns are linear combinations of the pivot columns. THEOREM: rank(A)=rank(A^T). DEF. DIMENSION. THEOREM. Two independent sets, each of which span a subspace S of a vector space V, must have the same number of vectors. This unique number is called the DIMENSION of the subspace S.: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012) THEOREM: A set of nonzero pairwise orthogonal vectors is linearly independent.Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)Slides

Purpose of basis: General solution with a minimal number of terms. Definition: Basis == independence + span. Differential Equations: General solution and shortest answer. Examples: Find a basis from a general solution formula. The role of partial derivatives in extracting information 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? Repeat for y = c1 + c2 exp(x). Basis calculations. Last Frame Algorithm: Basis for a linear system AX=0. Differentiate X on t1, t2, ... to find the basis. Pivot Theorem applications. The solution space of AX=0 is a subspace S (kernel theorem). S=span(columns of A)=span(pivot columns of A) Subspace basis from a spanning set. Apply the pivot theorem to find a largest set of independent vectors, generating a basis. Redundant vectors. Apply the pivot theorem to expunge irrelevant information and simplify representation of an answer. Example. Last frame algorithm and the vector general solution. Example. Given general solution y = c1 exp(x) + c2 exp(-x), What's the basis? From fixed vectors and matrices we have learned that: dim=number of partial derivatives on invented symbols basis=answers to partials on invented symbols We guess c1, c2 to be the invented symbols. Then exp(x) and exp(-x) is the basis. Independence delayed. Example. The solution space W of y'' - y = 0 has dimension 2. S1={exp(x), exp(-x)} and S2={cosh(x), sinh(x)} are two independent sets in W. By the theory, W=span(S1)=span(S2). This information simply means that there are two equally valid general solution formulas: y = c1 exp(x) + c2 exp(-x), and y = d1 cosh(x) + d2 sinh(x) FACT. Every nontrivial subspace has an infinite number of bases.

The pivot theorem is Algorithm 2, section 4.5.: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012)Slides

How to represent functions as graphs and as infinitely long column vectors.: Functions as infinitely long column vectors (123.8 K, pdf, 22 Dec 2012) Function rules for add and scalar multiply. Independence tests using functions as the vectors. Sampling Test Wronskian Test Web References:Slides: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014)Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012)Slides: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)Slides: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)Manuscript

PARTIAL FRACTION THEORY. DEF. Partial fraction = constant/poly with one rootFailsafe Method of Sampling.Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2) Example: top=x-1, bottom=(x+1)(x^2+1) Example: top=x-1, bottom=(x+1)^2(x^2+1)^2 Maple assist with convert(top/bottom,parfrac,x);The method of atoms.Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2)Heaviside's cover-up method.Example: 1/(x^2+3x+2)=A/(x+1) + B/(x+2) BOOT CAMP Your job is to dig trenches and then fill them up again. Briefly, do all the examples above by yourself, then throw away the scratch paper. Become a partial fraction expert, better than electrical engineer Oliver Heaviside. PROBLEM 4.7-26. How to solve y''+10y'=0 for general solution y=c1 + c2 exp(-10x) Tools: method of quadrature and integrating factors.

Def. atom=x^n(base atom) base atom = 1, exp(ax), cos(bx), sin(bx), exp(ax) cos(bx), exp(ax) sin(bx) for nonzero real a and positive real b THEOREM. Euler solution atoms are independent. EXAMPLE. Show 1, x^2, x^9 are independent by 3 methods. EXAMPLE. Independence of 1, x^2, x^(3/2) by the Wronskian test. REVISIT PROBLEM 4.7-26. Equation y''+10y'=0 has general solution y=c1 + c2 exp(-10x) The Euler solution atoms for this example are 1 and exp(-10x). Differential equations like this one have general solution a linear combination of atoms.

Picard: Order n of a DE = dimension of the solution space. General solution = linear combination n independent atoms. Euler's theorem(s), an algorithm for solution atoms.ROW RANK=COL RANK PROOF by the PIVOT THEOREMrank(A)=rank(A^T). Theorem 3, section 4.5. PROOF: rank(A)=number of independent rows=number of pivots rank(A)=rank(rref(A))=number of consecutive identity columns in rref(A) =nunber of independent rows in A^T (by pivot theorem)=rank(A^T)ALGEBRAIC INDEPENDENCE TESTS: review and examplesSampling test. Application to x^2,exp(x) Wronskian test. Application to 1,x,x^2,x^3 Orthogonal vector test. Example: (1,1,0), (1,-1,0), ((0,0,1) Pivot theorem. Example: Find the independent columns in a matrix. Example: Find the maximum number of independent vectors in a list. BASIS. Definition of basis and span. Examples: Find a basis from a general solution formula. Bases and the pivot theorem. Example: Find a basis for the row vectors in a matrix. Example: Find a basis or the column vectors in a matrix. Equivalence of bases. Example: A subspace S contains vectors v1,v2 and also vectors w1,w2. When are both v1,v2 and w1,w2 bases for S? A computer test for equivalent bases (slides). Web References:: Orthogonality, CSB-inequality, Pythagorean identity (124.8 K, pdf, 03 Mar 2012)Slides: The pivot theorem, dimension, row rank = col rank (189.2 K, pdf, 03 Mar 2012)Slides: Matrix add, scalar multiply and matrix multiply (156.7 K, pdf, 03 Mar 2012)Slides: Vector space, subspace, independence (168.4 K, pdf, 17 Feb 2014)Slides: Vector space, Independence, Basis, Dimension, Rank (268.7 K, pdf, 22 Mar 2010)Manuscript

Lab 7. Details forthcoming in Week 7. Some problems from Lab 6 are due Thursday.

: Atoms, Euler's theorem, 7 examples (130.5 K, pdf, 26 Feb 2013)Slides: Base atom, atom, basis for linear DE (117.6 K, pdf, 03 Mar 2012) EXAMPLE. The equation y'' +10y'=0. Review. How to solve y'' + 10y' = 0, chapter 1 methods. Midterm 1 problem 1(d). Idea: Let v=x'(t) to get a first order DE in v and a quadrature equation x'(t)=v(t). Solve the first order DE by the linear integrating factor method. Then insert the answer into x'(t)=v(t) and continue to solve for x(t) by quadrature. Vector space of functions: solution space of a differential equation. A basis for the solution space of y'' + 10y'=0 is {1,exp(-10x)} PICARDS THEOREM The theorem of Picard and Lindelof for y'=f(x,y) has an extension for systems of equations, which applies to scalar higher order linear differential equations. THEOREM [Picard] A homogeneous nth order linear differential equation with continuous coefficients has a general solution written as a linear combination of n independent solutions. This means that the solution space of the differential equation has dimension n. PICARD FAILURE. Although Picard's structure theorem does not provide an algorithm for construction of independent solutions, the theorems of Euler do that. Combined, there is an easy path to finding a basis for the solution space of an nth order linear differential equation. EULER'S THEOREM for CONSTRUCTING a SOLUTION BASIS It says y=exp(rx) is a solution of ay'' + by' + cy = 0 <==> r is a root of the characteristic equation ar^2+br+c=0. REAL EXPONENTIALS: If the root r is real, then the exponential is a real solution. THEOREM. A real root r=a (positive, negative or zero) produces one Euler solution atom exp(ax). COMPLEX EXPONENTIALS: If a nonreal root r=a+ib occurs, a complex number, then there is a conjugate root a-ib. The pair of roots produce two real solutions from EULER'S FORMULA (a trig topic): exp(i theta) = cos(theta) + i sin(theta) Details to obtain the two solutions will be delayed. The answer is THEOREM. A conjugate root pair a+ib,a-ib produces two independent Euler solution atoms exp(ax) cos(bx), exp(ax) sin(bx). HIGHER MULTIPLICITY For roots of the characteristic equation of multiplicity greater than one, there is a correction to the answer obtained in the two theorems above: Multiply the answers from the theorems by powers of x until the number of Euler solution atoms produced equals the multiplicity. EXAMPLE: If r=3,3,3,3,3 (multiplicity 5), then multiply exp(3x) by 1, x, x^2, x^3, x^4 to obtain 5 Euler solution atoms. EXAMPLE: If r=5+3i,5+3i (multiplicity 2), then there are roots r=5-3i,5-3i, making 4 roots. Multiply the two Euler atoms exp(5x)cos(3x), exp(4x)sin(3x) by 1, x to obtain 4 Euler solution atoms. SHORTCUT: The characteristic equation can be synthetically formed from the differential equation ay''+by'+cy=0 by the formal replacement y ==> 1, y' ==> r, y'' ==> r^2. EULER'S SOLUTION ATOMS Leonhard Euler described a complete solution to finding n independent solutions in the special case when the coefficients are constant. The Euler solutions are calledSlidesatomsin these lectures. THE TERM ATOM. The termatomabbreviatesEuler solution atom of a linear differential equation. The main theorem says that the answer to a homogeneous constant coefficient linear differential equation of higher order is a linear combinations of atoms. EULER SOLUTION ATOMS for LINEAR DIFFERENTIAL EQUATIONS DEF. Base atoms are 1, exp(a x), cos(b x), sin(b x), exp(ax)cos(bx), exp(ax)sin(bx). DEF: atom = x^n (base atom) for n=0,1,2,... THEOREM. Euler solution atoms are independent. THEOREM. Solutions of constant-coefficient homogeneous differential equations are linear combinations of a complete set of Euler solution atoms. EXAMPLE. The equation y''+10y'=0 has characteristic equation r^2+10r=0 with roots r=0, r=-10. Then Euler's theorem says exp(0x) and exp(-10x) are solutions. By vector space dimension theory, the Euler solution atoms 1, exp(-10x) are a basis for the solution space of the differential equation. Then the general solution is y = a linear combination of the Euler solution atoms y = c1 (1) + c2 (exp(-10x)).