Linear Algebra

226 Chapter Three. Maps Between SpacesProof Clear.QED3.21 Example This is the first system, from the first chapter, on which weperformed Gauss’s Method.3x 3 = 9x 1 + 5x 2 − 2x 3 = 2(1/3)x 1 + 2x 2 = 3We can reduce it with matrix multiplication. Swap the first and third rows,⎛⎜0 0 1⎞ ⎛⎞ ⎛⎞0 0 3 9 1/3 2 0 3⎟ ⎜⎟ ⎜⎟⎝0 1 0⎠⎝ 1 5 −2 2⎠ = ⎝ 1 5 −2 2⎠1 0 0 1/3 2 0 3 0 0 3 9triple the first row,⎛⎜3 0 0⎞ ⎛⎞ ⎛1/3 2 0 3⎟ ⎜⎟ ⎜1 6 0 9⎞⎟⎝0 1 0⎠⎝ 1 5 −2 2⎠ = ⎝1 5 −2 2⎠0 0 1 0 0 3 9 0 0 3 9and then add −1 times the first row to the second.⎛ ⎞ ⎛1 0 0⎜ ⎟ ⎜1 6 0 9⎞ ⎛⎞1 6 0 9⎟ ⎜⎟⎝−1 1 0⎠⎝1 5 −2 2⎠ = ⎝0 −1 −2 −7⎠0 0 1 0 0 3 9 0 0 3 9Now back substitution will give the solution.3.22 Example Gauss-Jordan reduction works the same way. For the matrix endingthe prior example, first adjust the leading entries⎛⎞ ⎛1 0 0⎜⎟ ⎜1 6 0 9⎞ ⎛⎟ ⎜1 6 0 9⎞⎟⎝0 −1 0⎠⎝0 −1 −2 −7⎠ = ⎝0 1 2 7⎠0 0 1/3 0 0 3 9 0 0 1 3and to finish, clear the third column and then the second column.⎛ ⎞ ⎛ ⎞ ⎛⎞ ⎛1 −6 0 1 0 0 1 6 0 9⎜ ⎟ ⎜ ⎟ ⎜⎟ ⎜1 0 0 3⎞⎟⎝0 1 0⎠⎝0 1 −2⎠⎝0 1 2 7⎠ = ⎝0 1 0 1⎠0 0 1 0 0 1 0 0 1 3 0 0 1 33.23 Corollary For any matrix H there are elementary reduction matrices R 1 , . . . ,R r such that R r · R r−1 · · · R 1 · H is in reduced echelon form.Until now we have taken the point of view that our primary objects of studyare vector spaces and the maps between them, and have adopted matrices onlyfor computational convenience. This subsection show that this isn’t the wholestory. Understanding matrices operations by how the entries combine can beuseful also. In the rest of this book we shall continue to focus on maps as theprimary objects but we will be pragmatic — if the matrix point of view givessome clearer idea then we will go with it.