# Course Outline for Linear Algebra

**Catalog Description
**This course covers matrices, their properties, operations and applications
with emphasis on systems of

linear equations; vector spaces, linear independence, and bases of vector spaces; linear transformations,

kernels and ranges; determinants, their properties and applications; eigenvalues and eigenvectors; the

standard inner product on R3; the Gram-Schmidt process; diagonalization of symmetric matrices; and

real quadratic forms.

**Course Objectives: See attached.**

Prepared by: | Reviewed by: | |||

Prof. Robert Tuskey Dept. of Mathematics |
Prof. Linda Padilla Department Chairperson Date |
|||

Revised | 11/02 | |||

Revised | 11/98 | Revised | 02/92 | |

Revised | 08/96 | Revised | 11/91 | |

Revised | 10/93 | Revised | 11/89 |

Week |
Topic or Class Activity |

1 | Sets, Functions, Matrices |

2 | More matrices, solving systems of equations |

3 | Vector spaces |

4 | More on vector spaces, linear independence |

5 | Spanning sets, bases, and finite dimensional
vector spaces |

6 | Rank of a matrix, structure of solutions of a
system of equations |

7 | Determinants |

8 | More on determinants, dot products |

9 | Orthogonality, The Gram-Schmidt process |

10 | Linear Transformations and matrix representations |

11 | Operations on linear transformations, null space and range |

12 | Change of basis, more on matrix representation |

13 | Similar matrices, eignevalues and eigenvectors |

14 | Diagonalization and symmetric matrices |

15 | Applications |

**OBJECTIVES
**Upon completion of this course you will be able to:

1. | Define "set" and the related terminology. |

2. | Define "function" and the related terminology. |

3. | Explain what is meant by and be able to form the composition of functions. |

4. | Explain what is meant by a system of equations, a
solution of the system, a consistent system, an inconsistent system and a homogeneous system of equations. |

5. | Define "matrix." |

6. | Explain what is meant by an "m x n" matrix, a
"square matrix of order n," the "(i,j) entry" of a matrix and the "main diagonal" of a square matrix. |

7. | Define and determine the equality of two
matrices, the sum of two matrices, the difference of two matrices, the product of two matrices, and the product of the scaler and a matrix. |

8. | Use summation notation in the definition of
matrix multiplication and proof of certain matrix properties. |

9. | Define "transpose of a matrix" and find the transpose of a given matrix. |

10. | Make a formal or informal proof of various theorems concerning the above objects and operations. |

11. | State all of the algebraic properties of matrix operations as discussed in class. |

12. | Prove selected algebraic properties of matrix
operations as well as various theorems which are off shoots of these properties. |

13. | State what is meant by the zero-matrix, by a
diagonal matrix, a scalar matrix, and the identity matrix of order n.. |

14. | Define "upper triangular form" and "lower triangular form" for a matrix. |

15. | Define "singular" matrix, "nonsingular" matrix,
and "inverse" of a matrix and find the inverse of a given matrix when it arrives. |

16. | Prove various theorems concerning the objects mentioned in Objectives 13 - 15. |

17. | Explain the connection between singular and
nonsingular matrices to the solution of a system of equations. |

18. | Define "n by n" elementary matrices of type I, II, or III. |

19. | Prove selected theorems concerning the operation of elementary matrices on a given matrix. |

20. | Use elementary matrices to develop a technique for finding the inverse of a given matrix. |

21. | Explain what is meant by row-reduced echelon form
for a matrix and transform a given matrix into row-reduced echelon form. |

22. | Define the three elementary row operations on a matrix. |

23. | Explain what is meant by one matrix being row equivalent to a second matrix. |

24. | Prove various theorems concerning row equivalence and row-reduced echelon form. |

25. | Use matrix techniques discussed in class to solve systems of linear equations. |

26. | Define "real vector space" and explain the
significance of each of the components of the definition. |

27. | Give examples of a vector space. |

28. | Define "subspace of a vector space" and give examples. |

29. | Determine whether or not a given object is a vector space or subspace. |

30. | Use appropriate notation, work problems, and
prove selected theorems involving vector spaces and subspaces. |

31. | Define "linear combination" of a set of vectors. |

32. | State what is meant by a set of vectors "spanning" a vector space. |

33. | Explain what is meant by a linearly dependent or linearly independent set of vectors. |

34. | Define a "basis" for a vector space. |

35. | Explain what is meant by a nonzero vector space. |

36. | Define the dimension of a nonzero vector space. |

37. | Give examples, use appropriate notation, work
problems, and prove selected theorems concerning linear dependence and independence, bases, and dimensions of vector spaces. |

38. | Define “row space” and “column space” of an m by n matrix. |

39. | Explain what is meant by the row (column) rank of a matrix. |

40. | Discuss the structure of a linear system of equations. |

41. | Define the “determinant” of an n by n matrix and evaluate the determinant of a given matrix. |

42. | Discuss and prove the various properties of
determinants and use these properties to aid in solving problems involving determinants. |

43. | Define the "minor" of an element a_{ij}of a matrix A. |

44. | Define the "cofactor" of an element a_{ij}of a matrix
A. |

45. | Explain and preform the process of finding a determinant by cofactor expansion. |

46. | Define the “adjoint” of a matrix A and find the adjoint of a given matrix. |

47. | Use appropriate notation and prove selected
theorems which demonstrate the connection among the inverse of a matrix, the determinant of a matrix, and the adjoint of a matrix. |

48. | Apply determinants in other selected situations as discussed in class. |

49. | Define the dot product of two vectors and discuss and/or prove its properties. |

50. | State the Cauch-Schwarz inequality and the triangle inequnt by cofactor expansion. |

51. | Define the “distance” between two vectors and what are “orthogonal” vectors. |

52. | Explain what is meant by an orthogonal set of vectors and an orthonormal set of vectors. |

53. | Define and calculate the scalar projection and vector projection of one vector on another. |

54. | Use appropriate notation, work problems, and
prove selected theorems concerning inner products, the Cauch-Schwarz and triangle inequalities, distance and orthogonality. |

55. | Discuss and use the Gram-Schmidt Proality for vectors. |

56. | Define "linear transformation" of a vector space V into a vector space W. |

57. | State what is meant by the “null space” and “range” of a linear transformation. |

58. | Explain what is meant by the matrix representation of a linear transformation. |

59. | Find the matrix representation of a given linear transformation. |

60. | Define the "sum," "scaler multiple" and
"composition" of linear transformations and thereby define a vector space of linear transformations. |

61. | State what is meant by a vector space of matrices. |

62. | Explain the concept of a coordinate vector with respect to an ordered basis. |

63. | Find how coordinate vectors transform under a change of basis. |

64. | Define "similar matrices.” |

65. | Give examples, use appropriate notation, work
problems, and prove selected theorems concerning rank of a matrix, linear transformations, null spaces, ranges, vector spaces of linear transformations, and vector spaces of matrices. |

66. | Define "diagonalizable linear transformation" and give example space of matrices. |

67. | Define "eigenvalue" and "eigenvector" of a linear
transformation, give examples, and find the eigenvalues of eigenvectors of a given matrix. |

68. | State what is meant by the characteristic polynomial of a matrix. |

69. | Work problems based on the definitions mentioned
in objectives 64-68 and theorems based on those definitions. |

70. | Explain what is meant by a symmetric matrix and
skew symmetric matrix, and determine whether or not a given matrix is symmetric or skew symmetric. |

71. | Discuss and/or prove the theorems connecting diagonalization and symmetric matrices. |

72. | Define "Real Quadratic Form" and "equivalence" of real quadratic forms. |

73. | Explain what is meant by congruent matrices. |

74. | Use appropriate notation, work problems and prove selected theorems involving quadratic forms. |