数学网课代修 Competency List by Section

Competency List by Section

数学网课代修 Understand the relationships between the solutions of Ax=b and the null, row, column spaces of A Understand the relationships between

Chapter 1 数学网课代修

Section 1.1

• Terminology
  
  
  
  • Homogeneous linear equation
  
  
  • Linear system
  
  
  • Unknowns
  
  
  • Solution of a linear system
  
  
  • Augmented matrix
  
  
  
  


• Distinguish linear from nonlinear equations (1)


• Perform Elementary Row Operations (17)


• Determine if a linear system is consistent or inconsistent (19)

Section 1.2

• Terminology
  
  
  
  • Reduced row echelon form
  
  
  • Row echelon form
  
  
  • Leading one
  
  
  • Leading variable
  
  
  • Free variable
  
  
  
  


• Solve linear system of equations using Gaussian Elimination (5)


• Solve linear system of equations using Gauss-Jordan Elimination (9)

Section 1.3

• Terminology
  
  
  
  • Matrix
  
  
  • Linear combination of matrices
  
  
  • Matrix Transpose
  
  
  • Trace of Matrix
  
  
  
  


• Represent matrix product as linear combinations


• Represent linear systems in matrix form Ax=b

Section 1.4

• Terminology
  
  
  
  • Identity matrix
  
  
  • Determinant of a 2x2 matrix
  
  
  
  


• Understand Properties of Matrix Arithmetic


• Understand Properties of Zero Matrices


• Calculate the determinant of a 2x2 matrix


• Calculate the inverse of a 2x2 matrix

Section 1.5

• Terminology
  
  
  
  • Row equivalent
  
  
  • Elementary matrix
  
  
  • Equivalency Theorem
  
  
  
  


• Determine inverse of a matrix using elementary row operations


• Determine if a matrix is invertible

Section 1.6

• Terminology
  
  
  
  • Equivalency Theorem
  
  
  
  


• Solve linear systems using matrix inversion

Section 1.7

• Terminology
  
  
  
  • Diagonal matrix
  
  
  • Triangular matrix
  
  
  • Symmetric matrix
  
  
  
  


• Determine powers and inverses of diagonal and triangular matrices

Chapter 2  数学网课代修

Section 2.1

• Terminology
  
  
  
  • Determinant
  
  
  • Minor of an entry
  
  
  • Cofactor an entry
  
  
  
  


• Calculate determinants using cofactor expansion

Section 2.2

• Calculate determinants using row reduction

Section 2.3

• Terminology
  
  
  
  • Equivalency Theorem
  
  
  
  


• Determine invertability using determinants.

EXAM 1

Chapter 3

Section 3.1

• Terminology
  
  
  
  • Vector
  
  
  • Vector Component
  
  
  • Ordered n-tuple
  
  
  • N-space
  
  
  
  


• Be able to add, subtract and translate vectors and multiply vectors with scalars.

Section 3.2

• Terminology
  
  
  
  • Norm of a vector
  
  
  • Unit vector
  
  
  • Dot Product
  
  
  
  


• Be able to perform vector operations.


• Calculate dot product of two vectors

Section 3.3

• Terminology
  
  
  
  • Orthogonal vectors
  
  
  • Point-normal form of the equation of a plane
  
  
  
  


• Calculate orthogonal projections of vectors


• Use vectors to derive equations of lines and planes and solve basic geometric problems

Section 3.4

• Terminology
  
  
  
  • Parametric equations of lines and planes
  
  
  • Vector form of the equation of a plane
  
  
  
  


• Dot product Form of a Linear System


• The relationship between Ax=0 and Ax=b

Section 3.5 (Optional)

• Terminology
  
  
  
  • Cross Product
  
  
  • Orthogonal vectors
  
  
  
  


• Understand relationships between dot and cross product of two vectors


• Calculate determinant form of cross product


• Understand geometric interpretation of cross product


• Calculate scalar triple product

Chapter 4  数学网课代修

Section 4.1

• Terminology
  
  
  
  • Vector Space Axioms
  
  
  
  


• Be able to determine if a set in a vector space.

Section 4.2

• Terminology
  
  
  
  • Subspace
  
  
  • Linear combination of vectors
  
  
  • Span of Vectors
  
  
  
  


• Be able to determine solution spaces of homogeneous systems.

Section 4.3

• Terminology
  
  
  
  • Linearly dependent and independent
  
  
  • Wronskian of a set of functions
  
  
  • Span of Vectors
  
  
  
  


• Be able to determine if a set of vectors is linearly (in)dependent


• Be able to determine if a set of functions are linearly (in)dependent

Section 4.4

• Terminology
  
  
  
  • Basis of vector spaces
  
  
  • Coordinates of a vector relative to a basis
  
  
  
  


• Be able to determine if a set of vectors forms a basis


• Be able to determine coordinate vectors relative to a basis

Section 4.5

• Terminology
  
  
  
  • Dimension of a vector space
  
  
  • Fundamental theorems
  
  
  
  


• Be able to determine the dimensions of a vector space

Section 4.6

• Be able to determine the transition matrices for changing basis

Section 4.7

• Terminology
  
  
  
  • Row and column vectors
  
  
  • Null, row and column spaces
  
  
  • Particular solution
  
  
  • General solution
  
  
  
  


• Understand the relationships between the solutions of Ax=b and the null, row, column spaces of A

Section 4.8

• Terminology
  
  
  
  • Rank of a matrix
  
  
  • Nullity of a matrix
  
  
  • Equivalent Statements
  
  
  • Over and underdetermined systems
  
  
  
  


• Understand the relationships between the solutions of a linear system and its dimensions

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