Having Solution or Not Hungyi Lee Reference Textbook

Having Solution or Not 李宏毅 Hung-yi Lee

Reference • Textbook: Chapter 1. 6

Learning Target Review x Linear System b System of Linear Equations Matrix-vector product: • Given A and b, sometimes x exists, and sometimes doesn’t. • A system of linear equations has solution or not. • New terms: “linear combination” and “span”

Solution • A system of linear equations is called consistent if it has one or more solutions. • A system of linear equations is called inconsistent if its solution set is empty (no solution). Solution set Consistent or Inconsistent ?

Solution (High School) More Variables ? • Considering any system of linear equations with 2 variables and 2 equations …… line 1 …… line 2 1 ine 1 =line 2 1 ine 1 1 ine 2 unique solution no solution infinitely many solution

Linear Combination

Linear Combination • What is the result of linear combination?

Column Aspect Vector set coefficients Linear Combination

System of Linear Equations v. s. Linear Combination Non empty solution set? Has solution or not? (A system of linear equations) Column Aspect Consistent? The Same question

Example 1 Has solution or not?

Example 1 Has solution or not? • No The linear combination is always on the dotted line.

Example 2 Has solution or not?

Example 2 Has solution or not?

Example 2 • If u and v are any nonparallel vectors in R 2, then every vector in R 2 is a linear combination of u and v • Nonparallel: u and v are nonzero vectors, and u cv. u and v are not parallel ? Has solution • If u, v and w are any nonparallel vectors in R 3, then every vector in R 3 is a linear combination of u, v and w?

Example 3 Has solution or not?

Example 3 • Has solution or not? Yes u and v are not parallel Has solution

Span

Span •

Span •

Span • (Different number of vectors can generate the same space. )

Span • nonparallel vectors

Span •

Span z z y y x x

Has solution or not? The same question

Summary Does a system of linear equations have solution? YES Have solution NO No solution