Augmented Matrices PDF - NE 112 Linear Algebra

NE 112 Linear algebra for nanotechnology engineering 6.4.3 Augmented matrices Douglas Wilhelm Harder, LEL, M.Math. [email protected] [email protected]  Augmented matrices...

NE 112 Linear algebra for nanotechnology engineering 6.4.3 Augmented matrices Douglas Wilhelm Harder, LEL, M.Math. [email protected] [email protected]  Augmented matrices Introduction In this topic, we will – Define augmented matrices – Show how operations on systems of linear equations have a corresponding operation on the augmented matrix – Discuss the benefits of using augmented matrices versus systems of linear equations 2  Augmented matrices Augmented matrices Let us start with either a system of linear equations, or equivalently, a linear combination of vectors equaling a target vector: a1,1 x1 + a1,2 x2 + a1,3 x3 + + a1,n xn = b1 a2,1 x1 + a2,2 x2 + a2,3 x3 + + a2,n xn = b2 a3,1 x1 + a3,2 x2 + a3,3 x3 + + a3, n xn = b3  b1   a1, j  b  a   2  2, j  am ,1 x1 + am ,2 x2 + am ,3 x3 + + am ,n xn = bm b =  b3  a j =  a3, j          x1a1 + x2a 2 + x3a3 + + xna n = b b  a   m  m, j  3  Augmented matrices Matrix representation Given the system of linear equations, we defined the matrix A – Let us augment the matrix by appending the vector b:  a1,1 a1,2 a1,3 a1,n  The dotted line a  separates the  2,1 a2,2 a2,3 a2,n  coefficients and A =  a3,1 a3,2 a3,3 a3,n  the right-hand   values   a am ,2 am ,3 am ,n   m ,1  a1,1 a1,2 a1,3 a1,n b1  a a2,2 a2,3 a2,n b2   2,1  ( A b ) =  a3,1 a3,2 a3,3 a3,n b3      a am ,2 am ,3 am ,n bm  4  m ,1  Augmented matrices Matrix representation While matrices will have a significance beyond the coefficients, the only purpose of the augmented matrix is to help us systematically find a solution to our problem – It is a data structure, not a mathematical object  a1,1 a1,2 a1,3 a1,n b1  a a2,2 a2,3 a2,n b2   2,1  (A b ) = ( a1 a 2 a3 an b ) =  a3,1 a3,2 a3,3 a3,n b3      a am ,2 am ,3 am ,n bm   m ,1 5  Augmented matrices Matrix representation Suppose we have a system of linear equation we’re trying to solve: x1 + 3 x2 = −2 1 3 −2  4 −5  4 x1 + 7 x2 = −5  7 Adding –4 times Eqn 1 onto Eqn 2 yields: x1 + 3 x2 = −2 1 3 −2  0 −5 3  −5 x2 = 3  Instead, add –4 times Row 1 onto Row 2 6  Augmented matrices Matrix representation As a second example − x1 − 2 x2 + 3 x3 + 8 x4 = −3  −1 −2 3 8 −3   3 −4 −6 7  3 x1 − 4 x2 + 5 x3 − 6 x4 = 7  5 Adding 3 times Eqn 1 onto Eqn 2 yields: − x1 − 2 x2 + 3 x3 + 8 x4 = −3  −1 −2 3 8 −3   0 −10 −2  −10 x2 + 14 x3 + 18 x4 = −2  14 18 We could have added 3 times Row 1 onto Row 2 7  Augmented matrices Use of augmented matrices We will subsequently reinterpret all of the steps taken in finding a solution to a system of linear equations to manipulations of an augmented matrix to find a solution to the corresponding system of linear equations – Row operations – Backward substitution – Free variables This will allow us to program these algorithms into a computer 8  Augmented matrices Use of augmented matrices As a side note, if we find a solution the system of linear equations corresponding to this augmented matrix 1 3 −2  4 7 −5    −0.2  we get x =    −0.6  What this means is that 1  3   −2  −0.2   − 0.6  7  =  −5  4      9  Augmented matrices Summary Following this topic, you now – Understand how to get an augmented matrix – Know that we can perform equivalent operations to what we did with equations on the augmented matrix – Are aware that it will be easier to describe and program such an algorithm without the variables – Understand that a solution to the system of linear equations gives the same solution to the question of finding a linear combination vectors equaling a target vector 10  Augmented matrices References https://en.wikipedia.org/wiki/Augmented_matrix 11  Augmented matrices Acknowledgments None so far. 12  Augmented matrices Colophon These slides were prepared using the Cambria typeface. Mathematical equations use Times New Roman, and source code is presented using Consolas. Mathematical equations are prepared in MathType by Design Science, Inc. Examples may be formulated and checked using Maple by Maplesoft, Inc. The photographs of flowers and a monarch butter appearing on the title slide and accenting the top of each other slide were taken at the Royal Botanical Gardens in October of 2017 by Douglas Wilhelm Harder. Please see https://www.rbg.ca/ for more information. 13  Augmented matrices Disclaimer These slides are provided for the NE 112 Linear algebra for nanotechnology engineering course taught at the University of Waterloo. The material in it reflects the authors’ best judgment in light of the information available to them at the time of preparation. Any reliance on these course slides by any party for any other purpose are the responsibility of such parties. The authors accept no responsibility for damages, if any, suffered by any party as a result of decisions made or actions based on these course slides for any other purpose than that for which it was intended. 14