Presentation is loading. Please wait. # A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:

## Presentation on theme: "A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:"— Presentation transcript:

A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore: Ax = b has a solution if and only if b is a linear combination of columns of A REVIEW

Theorem 4: The following statements are equivalent: 1.For each vector b, the equation has a solution. 2. Each vector b is a linear combination of the columns of A. 3. The columns of A span 4. A has a pivot position in every row. Note: Theorem 4 is about a coefficient matrix A, not an augmented matrix. REVIEW

1.5 Solution Sets of Linear Systems

Definition of Homogeneous A system of linear equations is said to be homogeneous if it can be written in the form Ax = 0, where A is an matrix and 0 is the zero vector in R m. Example: Note: Every homogeneous linear system is consistent. i.e. The homogeneous system Ax = 0 has at least one solution, namely the trivial solution, x = 0.

Important Question When does a homogenous system have a non-trivial solution? That is, when is there a non-zero vector x such that ?

Example 1: Determine if the following homogeneous system has a nontrivial solution: Geometrically, what does the solution set represent?

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable. Basic variables: The variables corresponding to pivot columns Free variables: he others

Example 2: Describe all solutions of the homogeneous system Geometrically, what does the solution set represent?

Example 3: Describe all solutions for Solutions of Nonhomogeneous Systems i.e. Describe all solutions of where and Geometrically, what does the solution set represent?

Homogeneous Nonhomogeneous

Homogeneous Nonhomogeneous x x y y zz

Theorem 6 Suppose is consistent for some given b, and let p be a solution. Then the solution set of is the set of all vectors of the form where is any solution of the homogeneous equation.

Download ppt "A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore:"

Similar presentations

Ads by Google