Review for Exam 1

The exam will cover all of the material in sections 1.1-2.2, excluding section 1.10.  Section 1.6 will only be covered under the "A"-level material.  In the following review, the material for the course is broken down into three levels: "C"-level, "B"-level, and "A"-level. Roughly, 70% of the points on the test will be based on the "C"-level material, 20% from the "B" level material, and 10% from the "A"-level

"C"-level material

(This section will make up 70% of the points on the exam).

Terminology:

We have learned many new terms this semester. Here are the top ten most important terms for the test:
  1. linear combination
  2. span
  3. linear independence
  4. parametric vector form
  5. one-to-one
  6. onto
  7. linear transformation
  8. superposition principle
  9. inverse of a matrix
  10. singular/nonsingular

Other Terms:  In addition to the above, you should also be familiar with the following: linear equation, coefficient, system of linear equations, solution set, row equivalence, equivalent systems, consistent vs. inconsistent, coefficient matrix, augmented matrix, elementary row operation, echelon form, reduced echelon form, basic variable, free variable, general solution, parametric description of solution set, back-substitution, column vector, vector equation, matrix equation, homogeneous, trivial/nontrivial solutions, transformation, domain, codomain, image, range, linear transformation, standard matrix for a linear transformation, diagonal entries, main diagonal, diagonal matrix, zero matrix, sums of matrices, matrix multiplication, powers of a matrix, transpose of a matrix,invertible, determinant (of a 2 by 2 matrix), elementary matrix, singular, nonsingular.

Fundamental Skills

  1. ROW REDUCTION: Make absolutely sure you can do this flawlessly. It comes up over and over again. We use row reduction to solve systems of linear equations, to solve vector equations, to solve matrix equations, to determine the solution set to a homogeneous system, to solve a nonhomogeneous system, to determine if a set of vectors is linearly independent, to find a vector which maps to a given vector under a matrix transformation, to determine whether a matrix transformation is one-to-one or onto, and to find the inverse of a matrix.
  2. Find the general solution to a system of equations, a vector equation, or a matrix equation.
  3. Write the solution to a linear system in parametric vector form.
  4. Determine when a set of vectors is linearly independent or show that it is linearly dependent.
  5. Determine if a given vector is in the span of a set of vectors.
  6. Find a vector whose image under a matrix transformation is a given vector.
  7. Find the standard matrix for a given linear transformation.
  8. Determine the transpose of two matrices.
  9. Multiply two matrices together.
  10. Calculate the determinant of a two by two matrix.
  11. Find the inverse of a two by two matrix.

"B"-level material

(This section will make up roughly 20% of the points on the exam). In general, the "B"-level material includes all of the "Essential" material described in the study guides. The following is a summary of the types of things you should be able to do (in addition to the "C"-level material) to get a B on the exam:

Key skills

  1. Determine whether a given matrix transformation is one-to-one or onto.
  2. Know the properties of matrix multiplication (in particular, know what is different from multiplication of real numbers).
  3. Construct an elementary matrix corresponding to any given elementary row operation.
  4. Calculate the inverse of a 3 by 3 matrix, using the algorithm on page 124.
  5. Determine when a matrix equation has more than just the "trivial" solution.
  6. Use properties of matrix multiplication to determine the transpose of the product of two matrices.
  7. Give a geometric description of the span of a set of vectors in R^3 (be careful to check whether they are linearly independent).
  8. Given a set of vectors where certain elements are given by unknown constants, determine values of those constants that would make the vectors linearly dependent.
(This section covers at most 10% of the points on the exam).

"A"-level material

The "A"-level material covers ALL the material assigned in the study guides, including the applications in section 1.6 . Additionally, you must be able to demonstrate a superior understanding of the theoretical material. For example, you should be able to do the following:

  1. Explain the relationship between matrix multiplication and composition of linear transformations.
  2. Answer True/False questions similar to the ones assigned in the study guides.
  3. Prove Theorem 8 on page 69.
  4. Use the definition of a linear transformation to show that T(0) =0.
  5. Explain why the standard matrix of a linear transformation is given by [T(e_1) T(e_2) ... T(e_n)].
  6. Show that the inverse of a square matrix, if it exists, is unique.