UNIT I. Systems of Linear Equations & Matrices
L01 Systems of Linear Equations; Matrices (Sections 1.1, 1.2)
LEARNING OBJECTIVES: Define a system of linear equations. Discuss the concepts of solution set and equivalent linear systems. Justify the number of solutions for a linear system. Define consistent and inconsistent systems. Introduce the matrix of coefficients and augmented matrix of a linear system. Solve the system by applying elementary row operations. Define leading entry, echelon form, and reduced echelon form. State the theorem on uniqueness of the reduced echelon matrix that equivalent to a given matrix.
TEXTBOOK HOMEWORK: 1.1 (# 1, 3, 5, 7, 11, 13, 15, 18, 19, 21)
1.2 (# 1, 2)
L02 Row Reduction and Echelon Forms (Section 1.2)
LEARNING OBJECTIVES: Define a pivot position, pivot column, and a pivot. Give a definition of the rank of a matrix. State step-by-step Row Reduction Algorithm. Use the Algorithms to change a matrix into a reduced echelon matrix. Solve a system of equations using the Row Reduction Algorithm. Define the basic variables and free variables of the reduced system of equation. If the system is consistent, give the solution set. State what necessary and sufficient conditions for a system to be consistent.
TEXTBOOK HOMEWORK: 1.2 (# 3, 7, 9, 11, 13, 15, 17, 19, 21, 22, 23, 24, 25, 26, 29, 31)
L03 Vectors & Vector Equations (Section 1.3)
LEARNING OBJECTIVES: Define column vectors in . Introduce vector operations. Give geometric description of column vectors and vector operations. Introduce vectors in . State properties of vector operations. Define linear combinations of vectors. Consider the question about a possibility of representing a given vector as a linear combination of other vectors. Observe that the answer to the question relates to existence of a solution of the corresponding linear system. Define . Give a geometric description of Span in the three-dimensional space. Solve application problems involving linear combinations.
TEXTBOOK HOMEWORK: 1.3 (# 1 – 15 (odd), 21, 23, 24, 25, 27)
L04 Matrix Equations (Section 1.4)
LEARNING OBJECTIVES: Consider a product of a matrix and a vector and describe it as linear combination of the columns of the matrix and the entries of the vector as weights. Represent a linear system in the form of a matrix equation Ax = b. State the Theorem that the Matrix Equation is equivalent to the corresponding Vector Equation and equivalent to the corresponding System of Linear Equations. State the four logically equivalent statements about existence of solution. Consider the question on existence of the solution to the matrix equation Ax = b for an arbitrary vector b. Give geometric interpretation of the Span of the columns of matrix A. State the Theorem that gives equivalent statements for the whole to be spanned by the vectors that are columns of an matrix A. Give a Row-Vector Rule for computing the product of a matrix and a vector,. Introduce the Identity matrix I. Prove the properties of the matrix-vector product.
TEXTBOOK HOMEWORK: 1.4 (# 1, 3, 5 – 9 (odd), 11, 13, 15, 16, 17, 19, 21, 25, 26, 27)
L05 Solution Sets of Linear Systems (Section 1.5)
LEARNING OBJECTIVES: Define a homogeneous linear system. Consider the question about existence of a nontrivial solution of the system. Write nontrivial solutions in parametric vector form. Consider the geometric meaning of the solutions sets for a homogeneous linear system. Discuss the way of representing the solution to the equation as a sum of the parametric vector solution to the homogeneous equation and a particular solution of the nonhomogeneous equation and give the geometric interpretation. Use the procedure of writing the solution set of a consistent system in parametric vector form.
TEXTBOOK HOMEWORK: 1.5 (# 1 – 23 (odd), 24)
L06 Applications of Linear Systems (Section 1.6)
LEARNING OBJECTIVES: Discuss Leontief “Exchange” Model for economics. Solve application problems on finding equilibrium prices for the economy (use MATLAB). Discuss the principles of “balancing” chemical equations. Solve problems on finding the whole numbers that would match the total number of each type of atoms on both sides of the equation. Define a network and network flow. State the basic assumptions on the network flow problems. Solve network problems.
TEXTBOOK HOMEWORK: 1.6 (# 1 – 9 (odd), 12, 13, 15)
L07 Linear Independence (Section 1.7)
LEARNING OBJECTIVES: Define a linearly independent set of vectors. State when a set is called linearly dependent and discuss linear dependence relation. Rephrase the definition of linear independence with respect to the column vectors of a matrix A. State the conditions of linear dependence (independence) of one vector, two vectors, two or more vectors. Explain why a set containing the zero vector is linear dependent. Give a geometric interpretation of linear dependence (independence) of sets of two and three vectors. Develop techniques of determining whether a set is linearly independent but inspection of the set where it is possible.
TEXTBOOK HOMEWORK: 1.7 (# 1 – 21 (odd), 22 – 32)
L08 Introduction to Linear Transformations (Sections 1.8, 1.9)
LEARNING OBJECTIVES: Define a transformation . Specify the domain and codomain of T. Define the image of a vector under the transformation T and the range of T. Give a definition of the matrix transformation and specify its range. Consider examples of the matrix transformations. Give a definition of a linear transformation. Consider properties of linear transformations. Introduce a standard matrix of a linear transformation. Discuss examples of contraction/dilation transformation and show that it’s a linear transformation. Consider examples of linear transformations in .
TEXTBOOK HOMEWORK: 1.8 (# 1 – 21 (odd), 22)
1.9 (# 1 – 21 (odd))
UNIT II. Determinants & Vector Spaces
L09 Existence and Uniqueness; Linear Models (Sections 1.9, 1.10)
LEARNING OBJECTIVES: Give definitions of “onto” and “one-to-one” transformations. Explain their meaning and give geometric interpretation. Determine whether a linear transformation is “onto” and/or “one-to-one”. Give necessary and sufficient conditions for the transformations to be “onto” and “one-to-one” by using the standard matrix of the transformation. Discuss an application problem on constructing a “Cambridge Diet” model. Discuss possible outcomes. Build and solve linear systems that arise in electrical network containing several loops. Solve problems on determining the loop currents in the given network.
TEXTBOOK HOMEWORK: 1.9 (# 23 – 30)
1.10 (# 1, 2, 5 – 8).
L10 Matrix Operations: Addition & Multiplication (Section 2.1)
LEARNING OBJECTIVES: Introduce matrix notation. Define some special matrices. Introduce operations of matrix addition and scalar multiplication. Discuss properties of the matrix operations. Define matrix multiplication as a composite linear transformation. Give two ways of calculating the product of two matrices: “matrix-column” rule and “row-column” rule. Use the ”row-column” rule to compute AB. Discuss properties of matrix multiplication. Be aware of some properties of multiplication, which hold for the numbers, but do not hold for the matrices. Consider examples illustrating it.
TEXTBOOK HOMEWORK: 2.1 (#1 – 12, 17)
L11 Transpose and Inverse of a Matrix (Section 2.1, 2.2)
LEARNING OBJECTIVES: Define powers of a matrix. Introduce the transpose of a matrix. Discuss properties of the transposes. Define the inverse of an invertible matrix. Give the formula for finding the inverse of a matrix A. Introduce for a matrix. Solve the problems on finding the inverse of a matrix. Tell how the invertible property of a matrix relates to solving the system Ax = b. State and prove properties of the inverses. Introduce the three types of elementary matrices. Discuss the theorem that gives an algorithm of finding the inverseof an invertible matrix A by using row operations. Outline the proof of the theorem bases on the elementary matrices. Discuss a way of finding the specified columns of the inverse without computing the whole matrix.
TEXTBOOK HOMEWORK: 2.1 (# 15, 16, 27, 28)
2.2 (# 1 – 13, 21 – 24, 29 – 36).
L12 Invertible Matrices; Determinants (Section 2.3, 3.1)
LEARNING OBJECTIVES: State and prove the Invertible Matrix Theorem. Discuss some corollaries of the Thorem. Define the inverse transformation. State the necessary and sufficient condition of a linear transformation to be invertible. Define the determinant on examples of square , , and, in general, matrices. Discuss the ways of computing the determinants for each. Give the properties of the determinants that directly follow from the definition.
TEXTBOOK HOMEWORK: 2.3 (# 1 – 7, 11 – 17, 33 – 39)
3.1 (# 1 – 7 (odd), 9 – 13 (odd), 15 – 18, 25 – 30, 39, 40)
L13 Properties of Determinants; Cramer’s Rule (Sections 3.2, 3.3)
LEARNING OBJECTIVES: State the Theorem3 (Row Operations and Determinants) in two forms. Outline a proof of the Theorem by using Method of Mathematical Induction. Consider the properties of determinants that follow from Theorem3 and the Invertible Matrix Theorem. Prove Multiplicative property of determinants. State and prove Cramer’s Rule for solving a linear system. Use Cramer’s Rule to solve systems of equations. Consider an application of Cramer’s rule to Laplace Transform.
TEXTBOOK HOMEWORK: 3.2 (# 1 – 4, 5 – 9 (odd), 11, 13, 15 – 25 (odd), 27 – 36, 37 – 43 (odd))
3.3 (# 1 – 9 (odd))
L14 Vector Spaces and Subspaces (Section 4.1)
LEARNING OBJECTIVES: Give a definition of a vector space. Consider examples of vector spaces – show that the ten axioms hold for them. Define a subspace of a vector space. Consider properties of subspaces. Give some examples of subspaces. Introduce a way of creating a subspace of V as a Span of a set of vectors in V. Give a geometric description of a span of two vector in . Give a definition of a spanning set for a subspace H. Determine whether a given set of vectors is a subspace and, then, find a spanning set, or show that the set is not a vector space. Determine whether a given vector is spanned by a given set. Describe the Span of a given set of vectors.
TEXTBOOK HOMEWORK: 4.1 (# 1 – 17, 20 – 24, 25 – 29)
L15 Null Spaces, Column Spaces, and Linear Transformations (Section 4.2)
LEARNING OBJECTIVES: Define the null space of a matrix . Prove the Theorem that a null space of an is a subspace of . Describe geometrical meaning of the null space of a matrix by means of the linear transformation. Discuss when the null space is the zero space. Determine whether a vector belongs to the null space of a matrix. Find the null space of a matrix. Describe a subspace of as a solution set of m homogeneous linear equations with n variables. Argue on the properties of Nul A. Discuss the difference between the implicit and explicit description of a null space. Tell how the linear independence of the columns of matrix relates to the null space of the matrix. Determine the spanning vectors of Nul A. Explain on an example why the spanning set for a null space is linearly independent. Define the column space of a matrix. Prove the theorem that the column space of an matrix is a subspace of . Discuss geometric meaning of the column space of a matrix. Determine whether a vector is in Col A. Explain why each column vector of matrix A belongs to Col A. Find the condition when Col A = . Relate the result to the linear transformation being “onto”. Introduce a liner transformation from one vector space to another. Define the kernel space and the range. Discuss the meaning of these concepts when a transformation is defined by a matrix.
TEXTBOOK HOMEWORK: 4.2 (# 1 – 13 (odd), 15 – 27)
L16 Linearly Independent Sets; Bases (4.3)
LEARNING OBJECTIVES: Give the definition of a linearly independent set of vectors. Indicate when a general set of vectors is linearly dependent. Give examples of linearly dependent sets. Define a basis of a subspace. Justify the statement that the columns of an invertible matrix form a basis in . Define the standard basis in . Define the standard basis in the set of polynomials of degree . Determine whether a set of vectors is a basis. State the Spanning set theorem that describes the procedure of building a basis from a set that spans a subspace. Describe the algorithm of finding a basis of the Nul A. Compare properties of the matrix A and its reduced echelon form B. Explain why the pivot columns of A form a basis for Col A. Find the bases of the Nul A and Col A.
TEXTBOOK HOMEWORK:
4.3 (# 1 – 15 (odd), 19 – 23 (odd), 26)
L17 Coordinates Systems (4.4)
LEARNING OBJECTIVES: Prove the theorem on the Unique Representation of a Vector in a space through a given basis. Define the coordinate vector of x relative to the basis B. Introduce the coordinate mapping determined by B. Consider a standard basis for a set of polynomials . Find x, given the coordinate vector of x relative a basis B. Show that , where E is the standard basis for . For a given x, find the coordinate vector relative to the basis B. Give a graphical interpretation of coordinates in the standard system and in B-system. Introduce the change-of-coordinate matrix from a basis B in into the standard basis in . Give its inverse. Justify why the coordinate mapping described by the change-of-coordinate matrix is a one-to-one linear transformation from onto . Find the change-of-coordinate matrix from a basis B to the standard basis and use it to calculate x, given , and vise verse.
TEXTBOOK HOMEWORK:
4.4 (# 1 – 12 (odd), 18)
Unit III. Subspaces, Eigenvalues & Inner Products
L18 Coordinate Mapping; Dimension of a Vector Space (4.4, 4.5)
LEARNING OBJECTIVES: Describe the coordinate mapping from a vector space V into . State the Theorem that the above transformation is a one-to-one linear transformation from V onto . Give a definition of an isomorphism from a vector space V onto a vector space W. Show that a plane in through the origin is isomorphic to . Consider some properties of polynomials by using the isomorphism between a subspace of the polynomials and . State the Theorem about the number of vectors in a basis of a vector space V. Define finite- and infinite-dimensional vector spaces. Give the dimensions of some spaces for which the bases are known. Find the dimension of a subspace spanned by a given set of vectors. Describe subspaces of . State and prove the Theorem about extending a linearly independent set of vector in a subspace H of a vector space V to a basis of H and compare the dimensions of the two. State and prove the Basis Theorem. Determine the dimensions of Nul A and Col A. Give an algorithm of generating a basis for V from a linearly independent subset of vectors in V.
TEXTBOOK HOMEWORK: 4.4 (#13, 15 – 16, 21, 27 – 31 (odd), 32)
4.5 (# 1 – 10, 11 – 19 (odd), 20, 21, 23, 29, 30, 33)
L19 Rank (4.6)
LEARNING OBJECTIVES: Define the row space of a matrix. Prove the Theorem that the row operations do not change the row space of a matrix. Give an algorithm of finding a basis for the row space of a matrix. Compare algorithms of finding bases for Row A, Col A, Nul A. Give a definition of the rank of a matrix. Consider the relations between the rank A, and dimensions of the Row A and Col A. Use the rank of a matrix to analyze existence of the solutions to the systems of nonhomogeneous equations. Complete the Inverse Matrix Theorem using concepts of basis, rank, and dimensions of Col A and Nul A. Give a geometric description of Nul A, Row A, and Col A if A is the standard matrix of a linear transformation.
TEXTBOOK HOMEWORK: 4.6 (# 1 – 16 (odd), 17 – 21, 27 – 30)
L20 Change of Basis (4.7)
LEARNING OBJECTIVES: State and prove the Theorem on existence and uniqueness of the change-of-coordinate matrix from one basis to another. Define the change-of-coordinate matrix from B to C. Find the change-of-coordinate matrix for a general vector space V with two bases, given the representation of the vectors in one basis through another. Find the coordinates of a vector with respect to a different basis. Explain the geometrical meaning of the transformation from to and vice versa. Consider bases in . Identify the change-of-coordinate matrix from a basis B in to the standard basis E in . Introduce two methods of evaluating the change-of-coordinate matrix from one basis to another in Find the basis given the change-of-coordinate matrix from this basis to the other given basis. Work with different bases in the sets of polynomials.
TEXTBOOK HOMEWORK: 4.7 (# 1 – 15 (odd), 16, 19)
L21 Eigenvectors and Eigenvalues (5.1)
LEARNING OBJECTIVES: Define an eigenvector of an matrix A. Give the geometric meaning of an eigenvector of a matrix A. Give a definition of an eigenvalue. Explain a relation between an eigenvalue and an eigenvector corresponding to the eigenvalue. Consider the definition of an eigenvalue and an eigenvector in terms of the matrix equation. Explain how you would determine whether a given scalar is an eigenvalue of a matrix. State what it actually means to find all eigenvectors corresponding to the given eigenvalue. Give a definition of the eigenspace of a matrix corresponding to a given eigenvalue. Tell how to find a basis for the corresponding eigenspace given the eigenvalue. State the theorem on the eigenvalues of a triangular matrix. Give the necessary and sufficient condition for to be an eigenvalue of a matrix A. State the theorem on linear independence of a set of eigenvectors corresponding to the distinct eigenvalues. Introduce the first-order difference equation that gives recursive description of a sequence in . Define a solution to a recursive equation, which gives an explicit description of the sequence. Depending on the initial value, discuss possible ways of constructing a solution to a recursive equation.
TEXTBOOK HOMEWORK: 5.1 (# 1 – 16 (odd), 17 – 24, 31 – 36)
L22 Evaluating Eigenvalues of a Matrix (5.1, 5.2)
LEARNING OBJECTIVES: Consider some properties of the eigenvalues of a matrix A. Give a continuation of the Invertible Matrix Theorem in relation to the eigenvalues and determinants. Introduce the characteristic equation of a matrix and the characteristic polynomial. Use the characteristic polynomial to find all eigenvalues of A and their multiplicities. Use the properties of the characteristic polynomial to get the total number of eigenvalues of a matrix counting their multiplicities. Give a definition of similar matrices. Consider properties of the similar matrices. Prove the theorem about the equality of the characteristic polynomials and eigenvalues for the similar matrices. Distinguish between similar matrices and row-equivalent matrices.
TEXTBOOK HOMEWORK: 5.1 (# 25 – 32)
5.2 (# 1 – 17 (odd), 18 – 22)
L23 Diagonalization (5.3)
LEARNING OBJECTIVES: Define a diagonalizable matrix. Consider powers of a diagonal matrix. Prove the formula on the powers of a diagonalizable matrix. Prove the Diagonalization Theorem. Develop an algorithm for diagonalization of a matrix. Consider a sufficient condition of a matrix to be diagonalizable. Consider matrices with multiple eigenvalues. Give the condition on the dimension of an eigenspace corresponding to an eigenvalue. Give necessary and sufficient conditions for a matrix to be diagonalizable. Explain how to construct an eigenvector basis for . Diagonalize a matrix if possible.
TEXTBOOK HOMEWORK: 5.3 (# 1 – 20 (odd), 21 – 28)
L24 Complex Eigenvalues (5.5)
LEARNING OBJECTIVES: Define an eigenvalue and eigenvector in the complex space . Analyze a relation between the complex eigenvalues and a matrix that defines a rotational transformation in . Discuss the properties of the complex eigenvalues and complex eigenvectors. Develop technique on finding the eigenvalues and a basis for the eigenspaces for a matrix when the eigenvalues are not real. Consider the rotation-scale matrix C. Define the angle of rotation and the scale factor . Relate each to the principal argument of the eigenvalue and modulus of the eigenvalue, respectively. Determine the angle of rotation and the scale factor for a transformation given by a matrix. Consider the theorem on similarity of a 2×2 real matrix, whose eigenvalues are , and a rotation-scale matrix.
TEXTBOOK HOMEWORK: 5.5 (# 1 – 20 (odd))
L25 Inner Product; Norm; Orthogonality
LEARNING OBJECTIVES: Define the inner product in Euclidean Space. Consider properties of the inner product. Define the norm of a vector. Explain its geometrical meaning. Prove the three properties of the norm. Define a unit vector in the direction of a given vector. Discuss the process of normalization. Introduce the distance between two vectors. Give its geometrical interpretation. Consider the geometrical point of view for two vectors to be orthogonal. Give an analytic definition of two vectors being orthogonal by using the inner product. State and prove the Pythagorean Theorem in Euclidean space. Explain the meaning of a vector to be orthogonal to a subspace. Define the orthogonal complement of a subspace. Consider two important properties of an orthogonal complement. State and prove the Theorem on orthogonality of Row A and Nul A for an matrix A. Consider the relation between Col A and Row of A-transposed to prove the second part of the theorem on orthogonality of Col A and Nul of A-transposed. Introduce the angle between two vectors in Euclidean space. State the geometric form of the inner product of nonzero vectors, which involves the angle between two vectors.
TEXTBOOK HOMEWORK: 6.1 (# 1 – 17 (odd), 19 – 31)
L26 Orthogonal Sets (6.2)
LEARNING OBJECTIVES: Define an orthogonal set of vectors. Prove that an orthogonal set of nonzero vectors is linearly independent. Introduce an orthogonal basis. State and prove the Decomposition Theorem through an orthogonal basis. For a given nonzero vector u,consider a problem of decomposing a vector into a sum of two orthogonal vectors: one is a scalar multiple of u and the other – orthogonal to u. Define each term in the indicated sum. Relate the orthogonal projection of a vector onto another vector to a projection onto a one-dimensional subspace spanned by the second vector. Compute the distance from a vector to a one-dimensional subspace. Give a geometric interpretation of the Decomposition Theorem for a p-dimensional subspace. Consider a problem of decomposing a force into the component forces.
TEXTBOOK HOMEWORK: 6.2 (# 1 – 16 (odd))
Unit IV. Orthogonalization & Linear Models
L27 Orthonormal Sets & Orthogonal Projections (6.2, 6.3)
LEARNING OBJECTIVES: Define an orthonormal set and an orthonormal basis. Determine if a set of vectors is an orthonormal basis. State necessary and sufficient conditions for an matrix to have orthonormal columns. State and prove three properties of the mapping , where U is a matrix with orthonormal columns. Give necessary and sufficient conditions for a square matrix to have orthonormal columns. Give a definition of an orthogonal matrix. Prove that an orthogonal matrix has both orthonormal columns and orthonormal rows. State and prove the Orthogonal Decomposition Theorem. Define the orthogonal projection of y onto a subspace W. Decompose y into a sum of its orthogonal projection onto W and the vector z in the orthogonal complement of W.
TEXTBOOK HOMEWORK: 6.2 (# 17 – 23 (odd), 24 – 30)
6.3 (# 1 – 10 (odd))
L28 Projections & Gram-Schmidt Process (6.3, 6.4)
LEARNING OBJECTIVES: State and prove the Best Approximation theorem. For a given vector y, define the closest point in a subspace W, the distance to W, and an error of approximation of y by elements of W. Prove that if , then the orthogonal projection of y onto W is the vector y itself. Calculate the closest point in a subspace W to y. Evaluate the distance from y to W. State and prove the properties of the orthogonal projection onto a subspace which possesses an orthonormal basis. Consider the relations expressed in terms of matrices that have orthonormal columns and, in particular, orthogonal matrices. Give a justification of the Gram-Schmidt process of producing an orthogonal basis for a subspace with a given basis. State the Gram-Schmidt Process theorem. Discuss purposes of scaling the vectors in an orthogonal basis. Explain why the Gram-Schmidt process is not suitable for computer-based calculations. Use the Gram-Schmidt process to orthogonalize a a given basis by hand computations.
TEXTBOOK HOMEWORK: 6.3 (# 11 – 15 (odd), 16 – 24)
6.4 (# 1 – 8, 9, 11)
L29 The Least-Squares Problems (6.5)
LEARNING OBJECTIVES: Introduce the Least-Squares Problem. Define a least-squares solution of Ax = b. Explain the geometric meaning of the general least-squares problem. Derive the normal equations for Ax = b. State the Theorem on the relation between the solutions of the normal equations and the least-squares solutions of Ax = b. Find the least-squares solutions of an inconsistent system. Give a definition of the least-squares error of the approximation. State and prove the theorem on three equivalent statements on uniqueness of the least-squares solution of Ax = b. Give an alternative way of calculating the least-squares solution of Ax = b, where A is a matrix with orthonormal columns.
TEXTBOOK HOMEWORK: 6.5 (# 1 – 13 (odd), 17, 18)
L30 Applications to Linear Models (6.6)
LEARNING OBJECTIVES: Introduce the notation used in statistical analysis to describe a linear model. Define the design matrix, the parameter vector, and the observation vector. Set up a system of equations for finding the parameter vector. Give a definition of the residual, the least-squares line (or regression line), and the regression coefficients. Justify why we end up solving the least-squares problem to find the line of regression. Discuss how the problem can be simplified if the data are put in mean-deviation form. Introduce the general linear models and a residual vector. Relate minimization of the length of a residual vector to finding the least-squares solution to the system. Give the normal equations for finding the least-squares solutions. Set up and solve a general linear problem.
TEXTBOOK HOMEWORK: 6.6 (#1 – 7)
L31 Diagonalization of Symmetric Matrices (7.1)
LEARNING OBJECTIVES: Define a symmetric matrix. Discuss simple properties of symmetric matrices. Determine whether a matrix is symmetric. Give a definition of an orthogonally diagonalizable matrix. Orthogonally diagonalize a matrix, if possible. Show that for a symmetric matrix, the eigenspaces are mutually orthogonal. State the criterion on a matrix to be orthogonally diagonalizable. Orthogonally diagonalize a matrix whose eigenvalue are not distinct. Define a spectrum of a matrix. State the Spectral Theorem for a symmetric matrix. Prove the Spectral Theorem.
TEXTBOOK HOMEWORK: 7.1 (# 1 – 23 (odd))
L32 Quadratic Forms (7.2)
LEARNING OBJECTIVES: Define a quadratic form on . Compute the quadratic form for a given matrix. Write the matrix of a quadratic form. Compute the value of a quadratic form for a given vector. Give a definition of a change of variable. Discuss the purpose of changing of variable in a quadratic form. State the principal axes theorem. Transform a quadratic form into a quadratic form with no cross-product term. Give a geometric view of the principal axes for an invertible 2×2 matrix. Graph an equation using the principal axes. Classify a quadratic form as positive definite, negative definite, or indefinite. Specify when a quadratic form is positive semidefinite or negative semidefinite. State and prove the Theorem on Quadratic forms and Eigenvalues. Give the geometric meaning of the positive definite, negative definite, or indefinite quadratic forms. Solve the problems on the conditional extremum for quadratic form using the eigenvalues & eigenvectors technique.
TEXTBOOK HOMEWORK: 7.2 (# 1 – 14, 19, 20)