Radnor High School - Radnor Township School District



Radnor High School

Course Syllabus

Linear Algebra Honors

0480

Credits:1 Grades: 11, 12

Weighted: Yes Prerequisite: AP Calculus AB, AP Calculus BC or

Length: Year teacher recommendation

Format: Meets daily

|Overall Description of Course |

|This college level course is designed to prepare the student for eventual courses in multivariable calculus and modern algebra. Students will |

|study systems of equations, vectors and vector spaces, linear transformations and matrix representations, determinants, eigenvectors and |

|eigenvalues and a variety of |

|applications. Linear algebra is used in abstract algebra, functional analysis and has extensive applications to both natural sciences and |

|social sciences. This course is an alternate year course. It will run in 2011‐12, but not in 2012‐13. This decision is subject to change based|

|on interest and potential enrollment. |

| |

|This course is intended for the highly motivated math students and designed to challenge the most mathematically capable students. The courses|

|will involve rigorous pacing and workload with teacher expectations intended to challenge the student. The course will require more |

|independent and self guided learning (with an emphasis on writing explanations) than all other courses. |

MARKING PERIOD ONE

|Common Core Standards |

|N-VM.1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use|

|appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v). |

|N-VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. |

|N-VM.4. (+) Add and subtract vectors. |

|N-VM.5. (+) Multiply a vector by a scalar. |

|N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. |

|N-VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. |

|N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions. |

|N-VM.9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but |

|still satisfies the associative and distributive properties. |

|N-VM.10. (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 |

|in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. |

|N-VM.11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with |

|matrices as transformations of vectors. |

|N-VM.12. (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.|

|A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. |

|A-REI.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of |

|dimension 3 × 3 or greater). |

|Student Objectives: |

|At the end of this quarter, student should be able to: |

|Solve a system of linear equations using Gaussian elimination |

|Add, subtract and multiply two matrices and perform scalar multiplication |

|Transpose a matrix and use properties of transposes |

|Find elementary matrices |

|Find the inverse of invertible matrices |

|Find the determinant of square matrices by cofactor expansion and row reduction |

|Use the properties of determinant |

|Perform operations on vectors in 2 and 3-spaces |

|Perform dot and cross products on vectors |

|Find the projection of a vector onto another vector |

|Write equation of lines and planes in 3-space |

|Materials & Texts |

| |

|TEXTS |

|Elementary Linear Algebra (Applications Version), 9th Ed. – Anton, H. & Rorres, C. |

|Activities, Assignments, & Assessments |

|ACTIVITIES |

| |

|Systems of Linear Equations and Matrices |

|Introduction to Systems of Linear Equations |

|Gaussian Elimination |

|Matrices and Matrix Operations |

|Inverses; Rules of Matrix Arithmetic |

|Elementary Matrices and a Method for Finding A-1 |

|Further Results on Systems of Equations and Invertibility |

|Diagonal, Triangular, and Symmetric Matrices |

| |

|Determinants |

|Determinants by Cofactor Expansion |

|Evaluating Determinants by Row Reduction |

|Properties of the Determinant Function |

| |

|Vectors in 2-Space and 3-Space |

|Introduction to Vectors (Geometric) |

|Norm of a Vector; Vector Arithmetic |

|Dot Product; Projections |

|Cross Product |

|Lines and Planes in 3-Space |

| |

|ASSIGNMENTS |

|Assignments #1 – 27 will be assigned. See the last page for the list of assignments. |

| |

|ASSESSMENTS |

|Homework will be assigned on a daily basis. Grades will be based on homework checks and chapter tests. |

|Terminology |

|Gaussian elimination, row reduction, reduced row-echelon form, transpose of a matrix, inverse of a matrix, trace of a matrix, singular matrix,|

|elementary matrix, symmetric matrix, skew-symmetric matrix, cofactor, minor of an entry, determinant, norm of a vector, dot product, cross |

|product, orthogonal projection of a vector, unit vector, scalar triple product, the normal of a plane |

|Media, Technology, Web Resources |

|Graphing calculator TI-84 or 89 |

MARKING PERIOD TWO

|Common Core Standards |

|N/A |

|Student Objectives: |

|At the end of this quarter, student should be able to: |

|Perform operations on vectors in the Euclidean n-space |

|Perform linear transformations from Rn to Rm |

|Determine if a set is a vector space |

|Determine if a set is a subspace |

|Determine if vectors are linearly independent |

|Find the basis and dimension of a vector space |

|Find the row space, column space and nullspace of a matrix |

|Find the rank and nullity of a matrix |

|Determine if a vector space is an inner product space |

|Find the inner product of elements in an inner product space |

|Find the angle between two vectors in an inner product space |

|Find the orthogonal complement of a subspace |

|Find the orthonormal basis using Gram-Schmidt process |

|Perform QR-decomposition |

|Find the new basis from an old basis |

|Determine if a matrix is orthogonal |

|Materials & Texts |

| |

|TEXTS |

|Elementary Linear Algebra (Applications Version), 9th Ed. – Anton, H. & Rorres, C. |

|Activities, Assignments, & Assessments |

|ACTIVITIES |

| |

|Euclidean Vector Spaces |

|Euclidean n-Space |

|Linear Transformations from Rn to Rm |

|Properties of Linear Transformations from Rn to Rm |

| |

|General Vector Spaces |

|Real Vector Spaces |

|Subspaces |

|Linear Independence |

|Basis and Dimension |

|Row Space, Column Space, and Nullspace |

|Rank and Nullity |

| |

|Inner Product Spaces |

|Inner Products |

|Angle and Orthogonality in Inner Product Spaces |

|Orthonormal Bases: Gram-Schmidt Process; QR-Decomposition |

|Best Approximation; Least Squares |

|Change of Basis |

|Orthogonal Matrices |

| |

|ASSIGNMENTS |

|Assignments #28 – 53 will be assigned. See the last page for the list of assignments. |

| |

|ASSESSMENTS |

|Homework will be assigned on a daily basis. Grades will be based on homework checks and chapter tests. A project and midterm exam will be |

|given. |

|Terminology |

|Vector space, Euclidean space, orthogonal vectors, linear transformation, linear operator, one-to-one, span, basis, linear independence, |

|basis, dimension, standard basis, row space, column space, nullspace, rank, nullity, inner product, orthogonal set, orthonormal set, |

|Gram-Schmidt process, QR-decomposition, |

|Media, Technology, Web Resources |

|Graphing calculator TI-84 or 89 |

MARKING PERIOD THREE

|Common Core Standards |

|N/A |

|Student Objectives: |

|At the end of this quarter, student should be able to: |

|Find the eigenvalues and eigenvector of square matrices |

|Diagonalize a square matrix |

|Find the power of a square matrix |

|Perform orthogonal diagonalization on a square matrix |

|Perform linear transformations |

|Determine the linear transformations for different vector spaces |

|Find the kernel and range of a linear transformation |

|Find the inverse of a linear transformation |

|Find the matrix of a general linear transformation |

|Determine if two matrices are similar |

|Determine the isomorphism between two vector spaces |

|Perform the LU-decomposition |

|Materials & Texts |

| |

|TEXTS |

|Elementary Linear Algebra (Applications Version), 9th Ed. – Anton, H. & Rorres, C. |

|Activities, Assignments, & Assessments |

|ACTIVITIES |

| |

|Eigenvalues, Eigenvectors |

|Eigenvalues and Eigenvectors |

|Diagonalization |

|Orthogonal Diagonalization |

| |

|Linear Transformations |

|General Linear Transformations |

|Kernel and range |

|Inverse Linear Transformations |

|Matrices of General Linear Transformations |

|Similarity |

|Isomorphism |

|LU-Decomposition |

| |

|ASSIGNMENTS |

|Assignments #54 – 63 will be assigned. See the last page for the list of assignments. |

| |

|ASSESSMENTS |

|Homework will be assigned on a daily basis. Grades will be based on homework checks and chapter tests. |

|Terminology |

|Eigenvalues, eigenvectors, diagonalization, orthogonal diagonalization, kernel, range, inverse linear transformation, similar matrices, |

|similarity invariant, isomorphism |

|Media, Technology, Web Resources |

|Graphing calculator TI-84 or 89 |

MARKING PERIOD FOUR

|Common Core Standards |

|N/A |

|Student Objectives: |

|At the end of this quarter, student should be able to: |

|Use mathematical induction to prove identities, inequalities and divisibility |

|Use the division algorithm and the Euclidean algorithm to find the greatest common divisor of two integers |

|Perform modulo operations |

|Use the properties of linear congruences |

|Use Fermat’s Little Theorem |

|Materials & Texts |

| |

|TEXTS |

|Elementary Linear Algebra (Applications Version), 9th Ed. – Anton, H. & Rorres, C. |

|Activities, Assignments, & Assessments |

|ACTIVITIES |

| |

|Additional topics from number theory. |

| |

|ASSIGNMENTS |

|Assignments #64 – 71 will be assigned. See the last page for the list of assignments. |

| |

|ASSESSMENTS |

|Homework will be assigned on a daily basis. Grades will be based on homework checks and chapter tests. A second project and a final exam will |

|also be given. |

|Terminology |

|Mathematical induction, division algorithm, Euclidean algorithm, greatest common divisor, least common multiple, linear congruences, modulo, |

|Fermat’s Little Theorem |

|Media, Technology, Web Resources |

|Graphing calculator TI-84 or 89 |

Linear Algebra Honors – Assignments

#1: Introduction to Systems of Linear Equations (1.1)

p. 6: # 1 – 7

#2: p. 7: # 8 – 14

#3: Gaussian Elimination (1.2)

p. 19 – 21: # 1 – 10, 12 – 14

#4: p. 21 – 22: # 17 – 25, 31

#5: Matrices and Matrix Operations (1.3)

p. 34 – 35: # 1 – 6, 7 – 10a, 11 – 14

#6: p. 35 – 37: #, 15a, 18 – 30

#7: Inverses; Rules of Matrix Arithmetic (1.4)

p. 48 - 50: # 1 – 12, 24 – 25,

#8: p. 49 – 50: # 13 – 17, 20 – 23

#9: Elementary Matrices and a Method for Finding A-1 (1.5)

p. 57 – 58: # 1 – 7, 8c & 8e

#10: p 58 – 59: # 10 – 13, 15, 22, 23

#11: Further Results on Systems of Equations and

Invertibility (1.6)

p. 66: # 1 – 16

#12: p. 67: # 20 – 27

#13: Diagonal, Triangular, Symmetric Matrices (1.7)

p. 73 – 74: # 1 – 8, 10

#14: p. 74 – 75: # 9, 12 – 18, 22

#15: Chapter 1 Review

p. 76 – 77: # 1 – 7, 9 – 11, 16, 19

#16: Determinants By Cofactor Expansion (2.1)

p. 94 – 95: #2a & b, 3b& c, 4, 7, 9, 11

#17: p. 95 – 96: #17, 22, 25, 27, 33, 35

#18: Evaluating Determinants By Row Reduction (2.2)

p. 101 – 102: # 3, 9, 12, 13, 19, 20

#19: Properties of the Determinant Function (2.3)

p. 109 – 111: #2, 3, 5, 7, 12a, 13, 18, 20, 22

#20: Chapter 2 Supplementary Exercises

p. 118 – 119: # 3, 6, 7, 8, 9, 15, 18b

#21: Introduction to Vectors (Geometric) (3.1)

p. 130 – 131: #1 (c – d), 2 (g – i), 3(e – f), 5, 6 (e-f),

9 – 11

#22: Norm of a Vector; Vector Arithmetic (3.2)

p. 134 – 135: #1 (d-f), 2 (c-d), 3 (d-f), 5, 10 – 12

#23: Dot Product; Projections (3.3)

p. 142 – 143: #3 (a-b), 4 (c-d), 6 (c – d), 12, 13, 25

#24: Cross Product (3.4)

p. 153 – 155: #1 c & d, 2a, 3c, 4b, 7, 8a, 10a, 11a

#25: p. 154 – 155: #20, 21, 24, 36, 37, 38

#26: Lines and Planes in 3-space (3.5)

p. 162 – 163: #1 – 15 (part a only)

#27: p. 163 – 164: #21, 22, 24, 40a, 42, 48

#28: Euclidean n-Space (4.1)

p. 178–179: #1 d & f, 3, 4, 5d, 9d, 10, 20

#29: p. 179 – 180: #22, 23, 25, 26, 27, 36

#30: Linear Transformation from Rn to Rm (4.2)

p. 193 – 194: #1, 2d, 4c, 6c, d, 8 – 11

#31: p. 194 – 195: #12 – 14, 18, 20, 26, 30, 31

#32: Properties of Linear Transformations from Rn

to Rm (4.3)

p. 206 – 207: #1, 2d, 4, 6c, 7-9

#33: p. 207 – 209: #14b, 15, 19a, 20, 21, 25

#34: Real Vector Spaces (5.1)

p. 226 – 227: #1, 6, 8, 9, 11

#35: p. 227 – 228: #12, 15, 17, 18, 19

#36: Subspaces (5.2)

p. 238 – 239: #1 – 6 (a – c)

#37: p. 239 – 240: #8a, 9a, 11, 12, 13, 21, 22

#38: Linear Independence (5.3)

p. 248 – 249: #1a & c, 3a, 4a, 5a, 7, 11

#39: p. 249 – 250: #13, 15, 19, 20a, 21b & c

#40: Basis and Dimension (5.4)

p. 263 – 264: ##1 - 6, 9a, 10b 

#41: p. 264 – 265: #13, 15, 19, 20, 21a, 23, 27a

#42: Row Space, Column Space, and Null Space (5.5)

p. 276 – 277: #2, 3 a , b, 5 c, 6 a, c & d, 7

#43: p. 278: #8 & 9 (use a & b of #6), 11 a, 12 a, 13, 15

#44: Rank and Nullity (5.6)

p. 288 – 289: #1, 2 b – d, 4, 5, 6, 11

#45: p. 289: #7, 8, 9, 10, 12a, 14, 15

#46: Inner Product (6.1)

p. 304 – 305: #3, 4, 9b & c, 10 a & b, 12, 13

#47: p. 305 – 306: #5, 7, 15a, 16 c & f, 17 – 20, 24, 27a

#48: Angle, Orthogonality in Inner Product Spaces (6.2)

p. 315 – 316: #2, 3, 5, 6a, 8b, 10, 11, 12a

#49: p. 315 – 316: #14 – 16, 18 b & c, 21, 22, 34

#50: Orthomormal Bases; Gram-Schmidt Process;

QR Decomposition (6.3)

p. 328 – 330: #2, 5a, 6b, 8, 10a, 11a, 12, 15

#51: p. 330 – 331: #17, 19, 21, 24a, c, 25, 31

#52: Change of Basis (6.5)

p. 345 – 346: #2, 3, 4, 8, 10, 12

#53: Orthogonal Matrices (6.6)

p. 354 – 355: #2, 3a – d, 4a, 14

#54: Eigenvalues & Eigenvectors (7.1)

p. 367–368: #5 & 6 (a, c, e of 4), 10, 12, 14, 15, 16,20

#55: Diagonalization (7.2)

p. 378 – 379: #1, 2, 7, 10, 11, 13, 21, 22, 24

#56: Orthogonal Diagonalization (7.3)

p. 383 – 385: #1, 4, 7, 10, 14. Supp.: #1, 3, 4, 6

#57: General Linear Transformations (8.1)

p. 398 – 399: #3 – 10, 14, 18, 22, 24, 31, 32

#58: Kernel & Range (8.2)

p. 405 – 406: #1 – 6, 10, 14, 15, 16, 18, 19, 26

#59: Inverse Linear Transformation (8.3)

p. 413– 415: #1 a – d, 3c, 4, 5, 7, 12, 16, 19, 20

#60: Matrices of General Linear Transformations (8.4)

p. 426 – 429: #2, 4, 6, 8, 9, 15, 18

#61: Similarity (8.5)

p. 439 – 441: #1, 3, 5, 8, 9a, 10, 12b, 15

#62: Isomorphism (8.6)

p. 445: #1 – 7, 8a, 9, 10

#63: LU-Decompositions (9.9)

p.518: # 3 – 11 odd, 12

Number Theory

#64: Mathematical Induction (1.1)

p. 6 – 7: #1 – 4, 6 – 10

#65: Division Algorithm & The Greatest Common

Divisor (2.1 & 2.2)

p. 25: #1 – 6, 13

#66: The Euclidean Algorithm & The Least Common

Multiple (2.3)

p. 31 – 32: #1, 2 a & b, 3, 5, 8, 12

#67: The Fundamental Theorem of Arithmetic (3.1)

p. 44: #1 – 5, 7, 8, 10

#68: Basic Properties of Congruence (4.2)

p. 68 – 69: #1 – 6

#69: Linear Congruences (4.4)

p.38: 1 – 3, 7; p. 82: #1 – 3

#70: More on Linear Congruences (4.4)

p. 82 – 83: 4, 5, 9, 10, 17, 19, 20

#71: Fermat’s Little Theorem (5.3)

p. 96 – 97: #1 – 4, 6, 7, 16

-----------------------

Created 6/21/2011

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download