1. 1.3 Solutions of Linear Systems

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

3. 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

4. If Rank = number of rows
How many solutions are possible?

5. If Rank = number of columns
0 4
0 3
How many solutions are possible?

6. 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)

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

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

9. 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

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

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

12. Problems 10, 12 and 14
Multiply the matrices if possible

13. 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)

14. 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.

15. Example 14 Ax =b

16. A definition and Problem 35

17. Homework p odd 34,46,47