1. How many solutions? Hung-yi Lee

2. Review No solution YES NO Given a system of linear equations with m equations and n variables Have solution We don’t know how many solutions

3. Today No solution YES Rank A = n Nullity A = 0 Unique solution Rank A < n Nullity A > 0 Infinite solution NO Given a system of linear equations with m equations and n variables Other cases?

4. How many solutions? Dependent and Independent

5. Dependent or Independent? Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

6. Dependent and Independent Dependent or Independent? Any set contains zero vector would be linear dependent Zero vector is the linear combination of any other vectors Flaw of the definition: How about a set with only one vector? Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

7. How to check? Given a vector set, {a 1, a 2, , a n }, there exists scalars x 1, x 2, , x n, that are not all zero, such that x 1 a 1 + x 2 a 2 +  + x n a n = 0. Given a vector set, {a 1, a 2, , a n }, if there exists any a i that is a linear combination of other vectors Linear Dependent

8. Another Definition How about the vector with only one element?

9. Intuition Intuitive link between dependence and the number of solutions Infinite Solution Dependent: Once we have solution, we have infinite. Dependent: Once we have solution, we have infinite.

10. Homogeneous Equations Homogeneous linear equations infinite

11. Homogeneous Equations Columns of A are dependent → If Ax=b have solution, it will have Infinite Solutions If Ax=b have Infinite solutions → Columns of A are dependent Non-zero

12. How many solutions? Rank and Nullity

13. Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank

14. Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank

15. Intuitive Definition The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank If A is a mxn matrix: Rank A = n Nullity A = 0 Columns of A are independent

16. How many solutions? Concluding Remarks

17. Conclusion No solution YES Rank A = n Nullity A = 0 Unique solution Rank A < n Nullity A > 0 Infinite solution NO

18. Conclusion Rank A = n Nullity A = 0 YES NO YESNO No solution Unique solution YESNO No solution Infinite solution

19. Question True or False If the columns of A are linear independent, then Ax=b has unique solution. If the columns of A are linear independent, then Ax=b has at most one solution. If the columns of A are linear dependent, then Ax=b has infinite solution. If the columns of A are linear independent, then Ax=0 (homogeneous equation) has unique solution. If the columns of A are linear dependent, then Ax=0 (homogeneous equation) has infinite solution.

20. Acknowledgement 感謝 姚鈞嚴 同學發現投影片上的錯誤