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MAS 3105 – LINEAR ALGEBRA

FLORIDA INT'L UNIV

REVISION FOR TEST #1

SPRING 2014

REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET KEY DEFINITIONS AND MAIN CONCEPTS: (Sections 1.1 - 1.5, 2.1 - 2.3, 3.1 - 3.3) Row & column vectors in Rn, inner product of two vectors & length of a vector, Linear equations, Systems of linear equations, Solution set, Equivalent systems, Consistent & inconsistent systems, Type I, II & III operations on equations, Gaussian elimination and b acksubstitution, Gauss-Jordan elimination, Homogeneous systems, Coefficient Matrix & Augmented Matrix of a system, Upper triangular & Lower triangular matrices, Row echelon form, Reduced row echelon form, Transpose of a matrix, scalar & Matrix multiplication, Diagonal, Identity & Null matrices; Non-singular matrices, Invertible matrices, the inverse of a matrix, Elementary matrices, Trace, definition of Determinant by using permutations, Minors & cofactors, Cofactor expansion along a row or along a column, , similar matrices, Vector spaces, Binary & scalar operation, Subspace, Linear combinations, Span of a set of vectors, Spanning set of a subspace, Linearly independent set of vectors.

MAIN PROBLEM SOVING TECHNIQUES OR ALGORITHMS: 1. 2. 3. 4. 5. 6. 7. 8.

Solving a system of linear equations: (a) by using Gaussian elimination & backsubstitution, (b) by using Gauss-Jordan (complete) elimination . Finding the inverse of a square matrix by using Type I, II & III row operations. Proving certain facts about matrices, their products, transposes, their inverses & products of these things and when row operations are done to the matrices.. Finding the determinant of a square matrix: (a) by using the cofactor expansion (b) by using row operations. Proving certain facts involving the determinants of the transposes, inverses & products of square matrices and when row operations are done to the underlying matrices. Determining whether or not a given set of vectors forms a subspace of a vector space. Determining if v is in span (v1, … , vk) , or if span (v1, … , vk) = Rn. Determining whether or not a given set of vectors is linearly independent.

MAIN THEOREMS & FORMULAS: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Theorem on the algebraic properties of matrices (Theorem 1.4.1 page 44). Theorem about the transpose of a matrix: (AT)T = A, (AB)T = BTAT. Theorem about the inverse of a matrix: (A-1)-1 = A, (AT)-1 = (A-1)T, (AB)-1 = B-1A-1. An nxn matix A is invertible if and only if A is non-singular. Any elementary matrix E is invertible and has an inverse of the same type Theorems about determinants when row operations are done to the underlying matrix. Theorems about trace & determinants : Tr (AB) = Tr(BA), det(A) = 0 iff A is singular, det (AB) = det(A) . det(B), det (AT) = det(A), det(A-1) = 1 / det(A). Theorem on the properties of vector spaces (Theorem 3.1.1, page 115). Theorem on uniqueness of a linear combination (Theorem 3.3.2 page 133).

MAS 3105 – LINEAR ALGEBRA

FLORIDA INT'L UNIV

REVISION FOR TEST #2

SPRING 2014

REMEMBER TO BRING AN 8x11 BLUE EXAM BOOKLET KEY DEFINITIONS AND MAIN CONCEPTS: (3.2 -3.6, 4.1- 4.3, 5.1-5.3, 5.5, 5.6) Subspace of a vector space, Linear combinations, Span of a set of vectors, Spanning set of a subspace, Linearly independent & linearly dependent set of vectors, Basis of a vector space, Finite-dimensional vector spaces, Dimension of a vector space, Standard basis of n, Coefficients of a vector with respect to an ordered basis, Row vectors and row space R(A) of a matrix A, Column vectors and column space C(A) of A, Rank of a matrix A, Null space N(A) of A, CoNull space CN(A) of A, Rank & Nullity theorem, Linear transformations, Kernel & Image of a linear transformation, Matrix representations of a linear transformation, Similarity& matrices representing the same linear transformation, Transition Matrix from one ordered basis to another ordered basis, Inner (dot) product in Rn, Angle between two vectors, Projections of a vector in the director of another vector, Vector equations of lines & planes, Orthogonal vectors & Orthogonal subspaces, Least squares Problems, Best least squares fit problems, Inner product spaces, Orthonormal sets & Orthonormal bases. ------ Eigenvalues (characteristic values), Eigenvectors, Trace of matrix, Diagonalizable matrices, Powers & Exponential of matrix A, Markov Chains. (Sections 6.1, 6.3). MAIN PROBLEM SOVING TECHNIQUES AND ALGORITHMS: 1. Determining whether or not a set of vectors span a given vector space or subspace. 2. Finding bases for R(A) & N(A) when A is an m by n matrix. Finding bases for CN(A) & C(A) when A is an m by n matrix. 3. Proving properties of linear transformations & proving a given set is a subpace. 4. Proving results about matrix representations of linear transf. and of similar matrices. 5. Finding the coefficient of a vector with respect to an ordered basis and finding the transition matrix from one ordered basis to another ordered basis. 6. Finding the least squares solution to an over-determined system of linear equations. 7. Finding the best least squares solution to a set of data by a linear function. 8. Finding an orthonormal bases from a given basis of n by the Gram-Schmidt process. ------ 9. Finding the eigenvalues & the corresponding eigenvectors of a square matrix A. 10. Finding a matrix Q which diagonalizes a given matrix A by using the eigenvectors of A.] MAIN 1. 2. 3. 4. 5. 6. 7.

THEOREMS & FORMULAS: Theorem on uniqueness of a linear combination (Theorem 3.3.2 page 133). If B is a basis of V with n elements, then all bases of V must also have n elements. dim [R(A)] = dim [C(A)], Rank(A) + Nullity(A) = Number of columns in A. If A is an m by n matrix, then [R(A)]T = N(A) , the subspace of n orthog. to N(A). If A is an m by n matrix, then [CN(A)]T = C(A), the subspace of m orthog. to C(A). Relationship between A & AT : C(AT) = [R(A)]T and N(AT) = [CN(A)]T If L is a linear transformation from V to W & S is a subspace of V, then ker(L) is a subspace of V and Im(L) & L[S] are subspaces of W. 8. A is similar to B iff A & B are different representations of the same linear transformation. ------ 9. Trace(A) = 1 + 2 + 3 + . . . + n , det(A) = 1 . 2 . 3 . . . . . n . 10. An n by n matrix A is diagonalizable iff A has n linearly independent eigenvectors.

MAS 3105 – LINEAR ALGEBRA INFORMATION SHEET (Jan. 3rd, 2014)

FLORIDA INT'L UNIV. SPRING 2014

INSTRUCTOR: OFFICE HOURS:

Prof. Ram Office: DM 419 E 1300-1350 Tue. & Thu. Office Tel.: 348-2929 1530-1640 Tue. (& maybe Thu.) Math Dept.: 348-2743 and at other mutually convenient times by appointment e-mail: [email protected] Webpage: http://faculty.fiu.edu/~ramsamuj/ PRE-REQUISITE: Calculus II or Discrete Mathematics (Grade C or better) A student needs a good working knowledge of algebraic manipulations (as used in Calculus II) and excellent arithmetical skills (without the use of a calculator) to succeed in this course. OFFICIAL TEXTBOOK: Linear Algebra with Applications (8th Ed.) by Steven J. Leon SYLLABUS: Below are the relevant sections for 95% of the course. 1. Matrices & Systems of Equations: Ch. 1 - Sec. 1, 2, 3, 4, 5 2. Determinants & their uses: Ch. 2 - Sec. 1, 2, 3* 3. Vector Spaces & their bases: Ch. 3 - Sec. 1, 2, 3, 4, 5, 6 4. Linear Transformations: Ch. 4 - Sec. 1, 2, 3 5. Inner products & Orthogonality: Ch. 5 - Sec. 1, 2, 3, 4*, 5, 6 6. Eigenvalues & Eigenvectors: Ch. 6 - Sec. 1, 3 The sections marked with an asterisk may be only partially covered. The other 5% (not in the text) is mostly about the co-null space of a matrix & representations of linear transformations. EXAM & ATTENDANCE POLICIES: 1. For each exam you will be required to bring your student ID & a blue exam booklet (SIZE: 8"x11" - available at the F.I.U. bookstore). 2. A make-up test for Exam #1 will be given only if there is a verifiable case of illness or emergency. If you miss Exam #2 for one of the same reasons that test will be discounted. 3. Religious holidays may be accommodated if I am notified in the first 2 week of classes. 4. Bathroom breaks will be allowed but you have to write in ink after returning. Any misconduct will be reported and dealt with according to the Code of Student Conduct. 5. Cell phones must be silenced (or set to vibrate) in class & are not allowed during exams. 6. Attendance is mandatory; if you attend less than 60% of the classes FIU will give you an “F”. SCHEDULE OF EXAMS: No calculators or formula sheets are allowed in the exams. Test #1 (100 points): THURSDAY, FEB. 20th, 1100-1215 Test #2 (100 points): THURSDAY, APR. 10th, 1100-1215 Final Exam (160 points): THURSDAY, APR. 24th, 0945-1145* * Note the earlier time. The final exam will be comprehensive. GRADING SCHEME: The grades will be assigned as indicated below. F | | 0% 48

D| 52

D | 56

D+ | 60

C| 64

C

C+ | 68

B| 72

B | 76

B+ | 80

A| 85

A | 90

HOLIDAYS: JAN. 20th (Mon.) - MLK, MAR. 10th-15th (Fri.-Sat.) - Spring Break. DEADLINE for DR or WI grade (no refund): MONDAY, MARCH 17th, 2014.

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