Drill:College algebra determinant definition Sarrus' rule for 2x2 and 3x3 matrices.The Four RulesTriangular rule [one-arrow Sarrus' rule] combo rule swap rule mult rule Examples: Computing det(A) easily. When does det(A)=0?

THEOREM. Determinant values for elementary matrices: det(E)=1 for combo(s,t,c), det(E)=m for mult(t,m), det(E)=-1 for swap(s,t).

Survey of Main theorems:Computation by the 4 rules, cofactor expansion, hybrid methods. Determinant product theorem det(AB)=det(A)det(B). Cramer's Rule for solving Ax=b:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta Adjugate formula: A adj(A) = adj(A) A = det(A) I Adjugate inverse formula inverse(A) = adjugate(A)/det(A).

REVIEWCofactor expansion of det(A). minor(A,i,j) checkerboard sign (-1)^{i+j} cofactor(A,i,j)=(sign)minor(A,i,j) Special theorems for det(A)=0 a zero row or col duplicates rows proportional rows. Elementary matrices Determinant product rule for an elementary matrix

Cofactor ExpansionExpansion details for 3x3 and 4x4. Hybrid methods to evaluate det(A). How to use the 4 rules to compute det(A) for any size matrix. Computing determinants of sizes 3x3, 4x4, 5x5 and higher. How to form the matrix of cofactors and its transpose. The adjugate matrix.

Frame sequences and determinantsFormula for det(A) in terms of swap and mult operations.

Cramer's ruleHow to solve Ax=b:

x_{1}= delta_{1}/delta, ... , x_{n}= delta_{n}/delta

- det(triangular matrix)=the product of the diagonal elements, and
- det(EA)=det(E)det(A), where E is an elementary combo, swap or mult matrix.

Determinant product theoremdet(AB)=det(A)det(B) for any two square matrices A,B Proof details. Example.

Adjugate matrix theoremsTHEOREM. The adjugate formula A adj(A) = adj(A) A = det(A) I.THEOREM. Adjugate inverse formula: inverse(A) = adj(A)/det(A).

Problem DetailsExercises 3.4-34 and 3.4-40. Cayley-Hamilton Theorem. It is a famous result in linear algebra which is the basis for solving systems of differential equations. Discussion of the Cayley-Hamilton theorem [Exercise 3.4-29; see also Section 6.3]: Determinants, Cramer's rule, Cayley-Hamilton (186.5 K, pdf, 09 Aug 2009)ManuscriptSuperposition proofProblem 3.4-40 is the superposition principle for the matrix equation Ax=b. It is the analog of the differential equation relation y=y_h + y_p. Web notes on the problems. Problem 3.4-29 is used in Problem 3.4-30. How to solve problem 3.4-30.

Four Vector Models:Fixed vectors Triad i,j,k algebraic calculus model Physics and Engineering arrows Gibbs vectors.: vector models and vector spaces (110.3 K, pdf, 03 Oct 2009) Parallelogram law. Head minus tail rule.SlidesVector ToolkitThe 8-property toolkit for vectors. Vector spaces. Reading: Section 4.1 in Edwards-Penney, especially the 8 properties.Lecture: Abstract vector spaces.Def: Vector==package of data items. Vectors are not arrows. The 8-Property Vector Toolkit Def: vector space, subspace Working set == subspace. Data set == Vector space Examples of vectors: Digital photos, Fourier coefficients, Taylor coefficients, Solutions to DE. Example: y=2exp(-x^2) for DE y'=-2xy, y(0)=2. RGB color separation and matrix add Intensity adjustments and scalar multiply

- Digital photos and matrix add, scalar multiply visualization.

Web references for chapter 4. Repeated below in ch3-ch4 references.

Sample Exam: Exam 1 key from F2009. See also S2009, exam 1.

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 page 239 or 243 section 4.2 of Edwards-Penney). 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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