STUDY GUIDE FOR MIDTERM EXAM 1

These are some of the main points that would be good to review for Midterm Exam 1. The sectionnumbers are taken from the textbook. It is important to understand definitions and examples givenin class and in recitation.

1. Chapter 2: Matrices and systems of linear equations

2.1: Matrices: Definitions and Notation.

Definition of a matrix. Row and column vectors. Square matrices. Upper-triangular and lower-triangular matrices. The transpose of a matrix. Symmetric and skew-symmetric matrices. Matrix functions.

2.2: Matrix Algebra.

Addition of matrices. Multiplication of a matrix by a scalar. Multiplication of matrices. Make sure to understand all the three cases we explained in class, i.e. row-vector times

column-vector, matrix times column-vector and matrix times matrix. Be sure to understand the sigma notation from Definition 2.2.12. Be careful to note that matrix multiplication is not commutative. Algebra and calculus of matrix functions.

2.3: Terminology for systems of linear equations.

Be able to identify the system coefficients (aij) and system constants (bj) of a system oflinear equations.

Understand the concept of a solution and the solution set of a system of linear equations. The matrix of coefficients and the augmented matrix associated to a linear system. Consistent and inconsistent systems. The picture for a 3x3 system. How many solutions can there be? How can we read them

off from the picture? Vector formulation of a linear system.

2.4: Elementary row operations and row-echelon matrices.

Elementary operations on systems of linear equations: permute equations, multiply anequation by a non-zero constant, add a multiple of one equation to another equation.

What does the above operation do to the augmented matrix? The elementary row operations on matrices: Pij ,Mi(k), Sij(k). Here, we always assumei 6= j and k 6= 0.

Row-equivalence of matrices. Understand why A B implies that B A. When can we do back-substitution in a linear system? Row-echelon matrices. Algorithm for reducing a matrix to row-echelon form (on Page 145). Understand the picture. The rank of a matrix. Reduced row-echelon form matrices.

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2 STUDY GUIDE FOR MIDTERM EXAM 1

2.5: Gaussian elimination.

Gaussian elimination. Gauss-Jordan elimination. When does a system of linear equations have a solution? When are there no solutions? (See

Lemma 2.5.3 and Lemma 2.5.5; you are not responsible for the proof in this class). Free and bound variables (parameters). How does one determine the number of free parameters from A and A#? (Lemma 2.5.7;

the statement only).

2.6: The inverse of a square matrix.

Invertible matrices vs. Singular matrices. Theorem 2.6.1. about the uniqueness of inverses. For this theorem, one should know

the proof. The inverse of a matrix (Definition 2.6.2). How does the knowledge of A1 help us solve A ~x = ~b? Invertibility and rank (statement of Theorem 2.6.5). Algorithm for computing the inverse of an invertible matrix; The Gauss-Jordan technique. Properties of the inverse; the statement of Theorem 2.6.9.

2. Chapter 3: Determinants

3.1: The definition of the determinant.

The determinant of a 1 1 and a 2 2 matrix. Permutations. Inversions. Odd and even permutations. The signature of a permutation. The determinant of an n n matrix. Definition 3.1.8. The explicit formula for the determinant of a 3 3 matrix (Sarrus rule). In other words,

in the 3 3 case, we can explicitly write the 6 terms in the sum without much calculation.It is important to note that this method doesnt apply to higher order determinants.

Geometric interpretation of the determinant: the area of a parallelogram and the volume ofa parallelepiped.

3.2: Properties of determinants.

The determinant of an upper/lower-triangular matrix. How do determinants change under elementary row operations? (Statements only). Computation of the determinant of a matrix by reducing this matrix to upper triangular

form and keeping track how the determinant changes at each step. Further properties of determinants: P4-P8 on Pages 203-204. (Statements only). Determinants and invertibility; statement of Theorem 3.2.4.

3. Chapter 4: Vector Spaces

4.2: Definition of a vector space.

Axioms for a vector space. Definition 4.2.1. The most important point for now is to understand whether the set we are considering is

closed under addition and under scalar multiplication, i.e. Axioms A1 and A2. Understand the examples and counterexamples from class. Examples of vector spaces on Page 248. Real and Complex vector spaces. Examples.