9/4/2008

Problem Session (Including Section 1.1 Exercises 6, 12, 27, and Section 1.2 Exercise 1)

Number of solutions of a linear system

A system of equations is said to be consistent if there is at least one solution; it is inconsistent if there are no solutions.

A linear system is inconsistent if and only if the reduced row-echelon form of its augmented matrix contains the row [ 0 0 ... 0 | 1], representing the equation 0 = 1.

If a linear system is consistent, then it has either

• infinitely many solutions (if there is at least one free variable), or
• exactly one solution (if all the variables are leading).

Rank of a matrix

The rank of a matrix A is the number of leading 1s in rref(A).

Systems with fewer equations than variables

A linear system with fewer equations than unknowns has either no solutions or infinitely many solutions.

Systems of n equations in n variables

A linear system of n equations in n variables has a unique solution if and only if the rank of its coefficient matrix A is n.

Matrix Algebra

• Sums of matrices
• Scalar multiples of matrices
• The identity matrix

The product Ax

• In terms of the columns of A
• In terms of the rows of A
• Algebraic rules

Linear combinations

Matrix form of a linear system

In class exercises

• Section 1.3 - 6, 9, 11, 19, 21, 29
• Chapter 1 Exercises - 3, 11, 15, 23, 35

Please read Sections 2.1 and 2.2 for class on Tuesday