Math 3191 <a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> Lecture 4: Matrix Equations and Solution Sets Stephen Billups University of Colorado at Denver Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.1/14 Announcements Study Guide 2 is posted. Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.2/14 Outline Matrix equations. Solution sets of linear systems Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.3/14 Sec. 1.4: Matrix Equations Key Concepts: Matrix-vector product. (last lecture) Dot product. (last lecture). Matrix equation. Relationships to vector equation, augmented matrix, and system of equations. Existence of solutions. Properties of Matrix-vector products. Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.4/14 Review Question Calculate the following Matrix-vector product: 3 1 2 -1 2 1 What about 2 1 5 -1 3 1 ? 3 2 4 Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.5/14 Matrix Equations A matrix equation has the form Ax = b. For example, 3 1 2 -1 x1 x2 = 4 1 . Key Fact: The equation Ax = b has a solution if and only if b is a linear combination of the columns of A. Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.6/14 Example 1 2 Let A = 2 3 . 1 1 Does the equation Ax = b have a solution for all b? Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.7/14 Relationship to Span Question: For what values of b will the matrix equation Ax = b have a solution? From previous slide, the equation is consistent if . This describes a plane in IR3 . Notice that this plane is exactly the span of the columns of A. Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.8/14 Theorem 4 Let A be an m n matrix. Then the following statements are logically equivalent: 1. For each b in IRm , the equation Ax = b has a solution. 2. Each b in IRm is a linear combination of the columns of A. 3. The columns of A span IRm . 4. A has a pivot position in every row. Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.9/14 Proof Outline We have already shown that the first three statements are equivalent. To complete the proof, we will show that statement 1. is equivalent to statement 4. First prove that if 4. is true, then 1. must be true. 1. Suppose 4. is true. Then for any b, row-reduce the augmented matrix [A b] to RREF, generating an equivalent augmented matrix [B d], where B is the RREF form of A. 2. Since A has a pivot position in every row, B has a pivot in every row, so there cannot be a pivot in the last column of the RREF of the augmented matrix. 3. Thus, the system is consistent for all right hand sides b. (i.e., 1. is true). Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >linear algebra</a> </a> p.10/14 Proof, continued Now prove that if 4. is false, then 1. must also be false. 1. If 4. is false, then the last row of B is all zeros. 2. If d is a vector with a 1 in the last entry, then the augmented matrix [B d] represents an inconsistent system. 3. Since B is row equivalent to A, we can apply a sequence of row operations to [B d] to produce a matrix [A b]. Where b is the vector produced by applying the row operations to d. 4. For this choice of b, the system Ax = b is inconsistent. (i.e., statement 1. is false). Math 3191<a href="/keyword/applied-linear-algebra/" >applied <a href="/keyword/linear-algebra/" >li...