MTH501 Final Term Mega File by GumNaaM Helpers

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TABLE OF CONTENTS :

Lesson 1: Introduction and Overview .................................................................................... 2 Lesson 2: Introduction to Matrices ...................................................................................... 10 Lesson 3: Systems of Linear Equations ............................................................................... 23 Lesson 4: Row Reduction and Echelon Forms .................................................................... 36 Lesson 5: Vector Equations ................................................................................................. 46 Lesson 6: Matrix Equations ................................................................................................. 62 Lesson 7: Solution Sets of Linear Systems .......................................................................... 74 Lesson 8: Linear Independence ........................................................................................... 84 Lesson 9: Linear Transformations ....................................................................................... 94 Lesson 10:The Matrix of a Linear Transformation ............................................................ 106 Lesson 11: Matrix Operations ............................................................................................ 116 Lesson 12: The Inverse of a Matrix ................................................................................... 129 Lesson 13: Characterizations of Invertible Matrices ......................................................... 143 Lesson 14: Partitioned Matrices......................................................................................... 157 Lesson 15: Matrix Factorizations....................................................................................... 175 Lesson 16: Iterative Solutions of Linear Systems .............................................................. 192 Lesson 17: Introduction to Determinant ............................................................................ 203 Lesson 18: Properties of Determinants .............................................................................. 219 Lesson 19: Cramer’s Rule, Volume, and Linear Transformations .................................... 234 Lesson 20: Vector Spaces and Subspaces .......................................................................... 245 Lesson 21: Null Spaces, Column Spaces, and Linear Transformations ............................ 269 Lesson 22: Linearly Independent Sets; Bases .................................................................... 292 Lesson 23: Coordinate System........................................................................................... 308 Lesson 24: Dimension of a Vector Space .......................................................................... 323 Lesson 25: Rank ................................................................................................................. 341 Lesson 26: Change of Basis ............................................................................................... 359 Lesson 27: Applications to Difference Equations ............................................................. 366 Lesson 28: Eigenvalues and Eigenvectors ......................................................................... 374 Lesson 29: The Characteristic Equation ............................................................................ 384 Lesson 30: Diagonalization................................................................................................ 396 Lesson 31: Eigenvectors and Linear Transformations ....................................................... 406 Lesson 32: Complex Eigenvalues ...................................................................................... 413 Lesson 33: Discrete Dynamical Systems ........................................................................... 419 Lesson 34: Applications to Differential Equations ............................................................ 429 Lesson 35: Iterative Estimates for Eigenvalues ................................................................. 442 Lesson 36: Revision ........................................................................................................... 451 Lesson 37: Revision ........................................................................................................... 452 Lesson 38: Inner Product ................................................................................................... 453 Lesson 39: Orthogonal and Orthonormal set ..................................................................... 460 Lesson 40: Orthogonal Decomposition ............................................................................. 477 Lesson 41: Orthogonal basis, Gram-Schmidt Process, Orthonormal basis ....................... 485 Lesson 42: Least Square Solution ...................................................................................... 493 Lesson 43: Inner Product Space ......................................................................................... 502 Lesson 44: Application of inner product spaces ................................................................ 512 Lesson 45: Revision ........................................................................................................... 523 22Linear Independence Sets; Bases VU

13. x+ y+ z = 0

3x+2y - 2z = 0

4x+3y - z = 0

6x+5y+ z = 0

Q14. Determine bases for the following subspace of R3

1. the plane 3x – 2y + 5z = 0
2. the plane x – y = 0
3. the line x = 2t, y = -t, z = 4t
4. all vectors of the form (a, b, c), where b = a + c
5. Find a standard basis vector that can be added to the set {v1, v2} to produce a basis

for R3. (a) v1 = (-1, 2, 3), v2 = (1, -2, -2) (b) v1 = (1, -1, 0), v2 = (3, 1, -2)

16. Find a standard basis vector that can be added to the set {v1, v2} to produce a basis

for R4. v1 = (1, -4, 2, -3), v2 = (-3, 8, -4, 6)