1.3 Solutions of Linear Systems

How many solutions does each of these systems have? Why? 4

Rank of matrix The rank of a matrix A is the number of leading 1’s in the rref (A) Determine the rank of the following matrices: 2 3 5 6 7 8 9

If Rank = number of rows 1 2 3 2 0 1 5 2 1 0 0 3 0 1 0 4 0 0 1 5 How many solutions are possible?

If Rank = number of columns 0 4 0 1 4 0 0 3 0 3 0 1 2 How many solutions are possible?

Implications of Rank Consider an mxn matrix (m rows , n columns) If rank A = m then the system is consistent (at least 1 sol.) If rank A = n then the system has at most 1 sol. If rank A < n then the system has either infinitely many solutions or no solutions. What if a does it mean about the solutions if a square matrix has full rank? (a nxn matrix with rank n)

More Implications of Rank If a linear system has fewer equations that unknowns then the system has either no solutions or infinitely many solutions

Adding matrices Add the components What types of matrices can be added? 5 –3 6 –6 1 2 3 –4 + a. 5 + 1 –3 + 2 6 + 3 –6 + (– 4) = 6 –1 9 –10 = 6 8 5 4 9 –1 – 1 –7 0 4 –2 3 b. 6 – 1 8 –(–7) 5 – 0 4 – 4 9 – (– 2) –1 – 3 = 5 15 5 0 11 –4 =

Scalar multiple of a matrix This is the scalar multiple of one column Scalar multiple of a matrix This is the scalar multiple of one column. If the matrix is larger just distribute the constant to all terms in the matrix

The product of a row vector with a column vector is a dot product

There are 5 ways to multiply matrices all which are important. We will learn 1 method now then the others in Chapter 2

Problems 10, 12 and 14 Multiply the matrices if possible

Which of the following matrices can be multiplied Which of the following matrices can be multiplied? (from the pre-Calc book 8.2, 37 and 38)

Linear combinations A vector b in Rn is called a linear combination of vectors v1,v2,v3, … vm if there exists scalars x1,x2,x3,…xm such that b = x1v1+ x2v2+x3v3….xmvm Note: this is the most fundamental operation in the course.

Example 14 Ax =b

A definition and Problem 35

Homework p. 33 1- 33 odd 34,46,47