MATH 262, Review for Midterm Test 1

Test topics

1. Linear systems (1.1)
2. Matrices, vectors and Gauss-Jordan elimination (1.2)
3. The number of solutions of a linear system. Row-reduced echelon form (1.3)
4. Matrix algebra (1.3)
5. Linear transformatitions. Matrix of a linear transformation (2.1)
6. Linear transformatitions in geometry (2.2)
7. Invertible transformations. The inverse of a matrix (2.3)
8. Matrix products (2.4)
9. Image and kernel of a linear transformation. Linear span. (3.1)

Objectives/Skills

• Write the system of linear equations in a matrix form and vice versa
• Write the matrix of a linear transformation
• Recognize basic types of 2 X 2 linear transformations (identity, scaling, reflection, projection, rotation, shear) from their matrices
• Solve linear systems using the Gauss-Jordan algortithm, including the case of infinitely many solutions (free variables)
• Understand the relation between vectors in linear algebra (columns) and geometric vectors. This includes parallel (collinear), perpendicular (orthogonal) vectors, angle between vectors, length of a vector, orthogonal projection
• Know the rules of matrix multiplication and understand their relation to composition of linear transformations
• Find inverses of matrices using the Gauss-Jordan algorithm and understand the relation between invertibility and rref
• Find the kernel and image given a matrix of a linear transformation

Formulas to memorize

• The Gauss-Jordan algorithms and the definition of rref and the rank
• Formula for the inverse of a 2 X 2 matrix (not required but helpful)
• Forms of matrices for projection, reflection, rotation (2 X 2)
• The rules of multiplication of matrices

Format of the test

Problems on the test will not be taken from the homework but they will often be similar. You will need to provide a complete solution for each problem, showing all steps. You may get partial credit for solutions that are not completely correct as long as you have the right ideas and explain them. The test will be closed books/notes. A simple scientific calculator (example: TI-30XII) is allowed, but it is not required to answer the questions. Graphing calculators (example: TI-83, 84, 89) may not be used. A number of questions on the test will be simple yes/no type questions, which may be taken from the review exercises in the end of each chapter.