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Course: MA 3191, Fall 2009
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Math 3191 &lt;a href=&quot;/keyword/applied-linear-algebra/&quot; &gt;applied &lt;a href=&quot;/keyword/linear-algebra/&quot; &gt;linear algebra&lt;/a&gt; &lt;/a&gt; Lecture 4: Matrix Equations and Solution Sets Stephen Billups University of Colorado at Denver Math 3191&lt;a href=&quot;/keyword/applied-linear-algebra/&quot; &gt;applied &lt;a...

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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...

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Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 5: Linear IndependenceStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/21AnnouncementsStudy Guide 3 and HWK 3 are posted. Discussion Page.Math 3191Applied Linear Alg
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Math 3191 Applied Linear AlgebraLecture 12: Null and Column SpacesStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/28AnnouncementsStudy Guide 6 posted HWK 6 postedMath 3191Applied Linear Algebra p.2/28S
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Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 14: Coordinate RepresentationsStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/22TonightFinish 4.3 Section 4.4: Coordinate RepresentationsMath 3191Applied Linear Alg
Colorado Denver - MA - 3191
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Colorado Denver - MA - 3191
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Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 18: Eigenvectors and EigenvaluesStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/25Section 5.1: Eigenvectors &amp; EigenvaluesDefinitions: Given an n n matrix A, if there
Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 19: DiagonalizationStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/19Section 5.3 DiagonalizationThe goal here is to develop a useful factorization A = P DP -1 , when
Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 20: Discrete Dynamical SystemsStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/17Sec. 5.4: Eigenvectors and Linear Transformations (cont.)Review: Last time, we looked
Colorado Denver - MA - 3191
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Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 23: Orthogonal Projections, Gram-SchmidtStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/26Orthonormal SetsA set of vectors {u1 , u2 , . . . , up } in Rn is called an
Colorado Denver - MA - 3191
Math 3191 Applied Linear AlgebraLecture 24: Gram-Schmidt and Least SquaresStephen Billups University of Colorado at DenverMath 3191Applied Linear Algebra p.1/21Section 6.4The Gram-Schmidt ProcessGoal: Form an orthogonal basis for a subspac
Colorado Denver - MA - 3191
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Math 5490 Network FlowsLecture 20: Min-Cost Flow ProblemsStephen Billups University of Colorado at DenverMath 5490Network Flows p.1/19PreliminariesAnnouncements: Sign-up for presentation times before Monday. (May 2nd and April 27th are taken)
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Colorado Denver - MA - 5594
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Colorado Denver - MA - 5594
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Colorado Denver - MA - 5594
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Colorado Denver - MA - 5594
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Colorado Denver - MA - 5594
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Colorado Denver - MA - 5610
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Colorado Denver - MA - 5610
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Colorado Denver - MA - 5610
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