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
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Created 6/21/2011
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