MATH 262, Review for Midterm Test 2

Test topics

1. Dimension Theory
   1. Linear Span of a Set of Vectors (3.2)
   2. Linear Independence (3.3)
   3. Basis (1.3)
   4. Dimension (1.4)
   5. Coordinates (1.5)

2. Systems of Linear Equations
   1. Matrix Algebra (5.1)
   2. Matrix Inverses (5.2)

3. Linearity
   1. Linear Functions (Transformations) (6.1)
   2. Compositions and Inverses (6.2)
   3. Matrix of a Linear Transformation (6.3)
   4. Matrices of Compositions and Inverses (6.4)
   5. Change of Basis (6.5)
   6. Image and Kernel (6.6)
   7. Rank and Nullity (6.7)

Key Concepts

• Linear combinations, linear independence, linear span of a set of vectors.
• Definition of basis. Comparison Theorem. Contraction Theorem. Expansion Theorem
• Dimension.
• Coordinates of a vector. Notation for column-vector of coordinates (brackets).
• Coordinate Transformation. Change of coordinates matrix.
• Matrix Multiplication.
• Matrix Inverse. Finding Inverse by Gaussian Elimination
• Linear and nonlinear functions.
• Matrix Multiplication corresponds to Composition of Linear Transformations.
• Matrix Inverse corresponds to the Inverse Transformation
• Image and Kernel. How to find Basis of a Kernel and a Basis of an Image
• Rank and Nullity. Dimension Theorem.

General rules

• Problems will come in the same format as on the quizzes: you will need to present a complete solution of the problem showing all steps to receive full credit.